Analytic Geometry Formulas
someonesdad1@gmail.com 5 Sep 2012
This file is AnalyticGeometry.pdf from http://code.google.com/p/hobbyutil/.
Revision history:
5 Sep 2012
Added mensuration, trigonometric identities, series, and constants sections (I'm
always looking these up, so I thought I'd put them here even though they're not
really analytic geometry). Other minor changes.
6 Feb 2011
Original release.
This document is a collection of formulas from analytic geometry and a few other things. Since
vectors are used for conciseness, there are also a few formulas from vector analysis.
If you find an error or obvious omission, please notify me at the above email address and it will get
fixed in a subsequent revision.
Document maintainer: make the Check paragraph style visible to see document maintenance
instructions and the details of the checks made on the formulas (but not all formulas have been
checked). Written in Open Office 3.2.1.
Table of Contents
Copyright.....................................................................................................................................5
Notation.......................................................................................................................................6
Constants....................................................................................................................................7
Series, approximations, and formulas.......................................................................................10
Trigonometry.............................................................................................................................12
Relations between the circular functions..............................................................................14
Magnitude and sign...............................................................................................................15
Angles in other quadrants.....................................................................................................15
Small angle approximations..................................................................................................15
Ordering relations.................................................................................................................15
Values for some special angles............................................................................................15
Pythagorean relationships....................................................................................................16
Sums and products...............................................................................................................16
Multiple angles......................................................................................................................17
Half angles............................................................................................................................17
Powers..................................................................................................................................17
Inverse functions...................................................................................................................18
Principal values.................................................................................................................18
Relations...........................................................................................................................18
Triangles...............................................................................................................................19
Solution of right triangles......................................................................................................21
Solution of oblique triangles..................................................................................................22
ASA...................................................................................................................................22
SSA...................................................................................................................................22
SAS...................................................................................................................................23
SSS...................................................................................................................................23
Mensuration...............................................................................................................................24
Two-dimensional figures.......................................................................................................25
Rectangle..........................................................................................................................25
Right triangle.....................................................................................................................25
Isosceles triangle..............................................................................................................26
General triangle................................................................................................................27
Parallelogram....................................................................................................................28
Trapezoid..........................................................................................................................28
Trapezium.........................................................................................................................29
Quadrilateral.....................................................................................................................29
General polygon...............................................................................................................30
Hexagon (regular).............................................................................................................31
Octagon (regular).............................................................................................................32
Polygon (regular)..............................................................................................................33
Circle.................................................................................................................................34
Circular sector...................................................................................................................34
Circular segment..............................................................................................................35
Annulus.............................................................................................................................38
Annular sector...................................................................................................................38
Spandrel...........................................................................................................................39
Ellipse...............................................................................................................................40
Parabola...........................................................................................................................42
Cycloid..............................................................................................................................42
Approximate area.............................................................................................................42
Three-dimensional figures....................................................................................................44
Cube.................................................................................................................................44
Rectangular prism............................................................................................................45
Cylinder.............................................................................................................................45
Hollow circular cylinder.....................................................................................................46
Truncated right circular cylinder.......................................................................................46
Pyramid or cone...............................................................................................................47
Tetrahedron......................................................................................................................48
Frustum of pyramid or cone..............................................................................................49
Regular polyhedra............................................................................................................49
Obelisk..............................................................................................................................49
Parallelepiped...................................................................................................................50
Prismoidal formula............................................................................................................51
Wedge..............................................................................................................................51
Sphere..............................................................................................................................52
Spherical sector................................................................................................................52
Spherical segment (spherical cap)...................................................................................53
Spherical zone..................................................................................................................53
Lune..................................................................................................................................53
Ellipsoid............................................................................................................................54
Spheroid...........................................................................................................................54
Paraboloidal segment.......................................................................................................54
Torus.................................................................................................................................55
Barrel................................................................................................................................56
Solid or surface of revolution................................................................................................56
Cavalieri's theorem...............................................................................................................58
Plane geometry.........................................................................................................................58
Determining a right triangle...................................................................................................58
Area of a triangle...................................................................................................................58
Inscribed circle of a triangle..................................................................................................58
Circumscribed circle of a triangle..........................................................................................58
Equation of a line..................................................................................................................59
Polar coordinates..............................................................................................................60
Angle between two lines.......................................................................................................60
Distance from a line to a point .............................................................................................61
Bisectors of angles between two lines..................................................................................61
Equation of circle..................................................................................................................61
Polar equation of circle.....................................................................................................61
Equation of parabola.............................................................................................................62
Equation of ellipse.................................................................................................................63
Equation of hyperbola...........................................................................................................65
Conic sections.......................................................................................................................66
Conic sections with matrix notation......................................................................................67
Circle through three points ...................................................................................................69
Tangent to a circle................................................................................................................69
Angle of intersection of two circles.......................................................................................71
Center of triangle's inscribed circle.......................................................................................71
Euclidean Transformations...................................................................................................71
Translations......................................................................................................................71
Rotations...........................................................................................................................72
Oblique transformations........................................................................................................72
Oblique coordinates..........................................................................................................72
Archimedean spiral...............................................................................................................73
Curvature..............................................................................................................................73
Products....................................................................................................................................74
Pseudovectors and pseudoscalars.......................................................................................74
Scalar (dot) product..............................................................................................................74
Vector (cross) product..........................................................................................................75
Scalar triple product..............................................................................................................76
Three noncoplanar vectors as a base (reciprocal system)..............................................77
Vector triple product..............................................................................................................77
Direction cosines.......................................................................................................................78
Equation of a line......................................................................................................................79
Forms....................................................................................................................................79
Vector...............................................................................................................................79
Endpoints of two vectors..................................................................................................79
Parametric........................................................................................................................79
Symmetric.........................................................................................................................79
Two-point form..................................................................................................................79
Equation from two points on line...........................................................................................79
From two planes...................................................................................................................80
Equation of a plane...................................................................................................................80
General form.........................................................................................................................80
Hessian normal form........................................................................................................80
Dual form..........................................................................................................................81
From axial intercepts.............................................................................................................81
Through origin and parallel to two vectors............................................................................81
From three noncollinear points.............................................................................................81
Through a point and perpendicular to a vector.....................................................................82
From two simultaneous equations: projecting planes.........................................................83
To find the projecting plane of a given line.......................................................................83
Points........................................................................................................................................83
Collinear points.....................................................................................................................83
Coplanar points.....................................................................................................................83
Lines and Planes.......................................................................................................................84
Point dividing a line segment into a given ratio....................................................................84
Lines in space.......................................................................................................................84
Line in a plane.......................................................................................................................84
Line through a point parallel to a vector...............................................................................84
Intersection of two lines........................................................................................................85
Intersection of two planes.....................................................................................................85
Intersection of three planes..................................................................................................85
Intersection of four planes....................................................................................................86
Intersection of a line and plane.............................................................................................86
Angle between two lines in space........................................................................................86
Angle between a line and a plane........................................................................................87
Angle between two planes....................................................................................................87
Parallel planes......................................................................................................................87
Perpendicular planes............................................................................................................87
Plane through a point parallel to two vectors........................................................................87
Plane through a point parallel to another plane....................................................................88
Plane through a line parallel to another line.........................................................................88
Plane through a given point and line....................................................................................88
Plane through a point and normal to a vector......................................................................88
Line perpendicular to a plane containing three points..........................................................88
Line perpendicular to two lines.............................................................................................89
Distance of a plane to the origin...........................................................................................89
Distance of a point to a line..................................................................................................89
Distance of a point from a plane...........................................................................................89
Distance between two non-intersecting lines.......................................................................89
Circle in space...........................................................................................................................90
Vectors......................................................................................................................................90
First degree vector equation.................................................................................................90
Invariance.............................................................................................................................91
Spheres.....................................................................................................................................91
Miscellaneous............................................................................................................................92
Center of mass......................................................................................................................92
Projections............................................................................................................................92
Isometric...........................................................................................................................92
Projection angles..............................................................................................................93
Sine sticks.........................................................................................................................94
Rotations...............................................................................................................................95
Rotating about a given axis..............................................................................................96
Direction cosine matrix to Euler angles............................................................................97
Direction cosine matrix to Euler axis and angle...............................................................97
Rotating lines in space..........................................................................................................97
Helix....................................................................................................................................100
Solid angles........................................................................................................................100
Layout......................................................................................................................................102
Cutting pipe ends for welding.............................................................................................102
Dividing a line equally.........................................................................................................104
Geometric mean.................................................................................................................105
Golden ratio........................................................................................................................106
Inscribed/circumscribed circle, etc......................................................................................106
Line segment perpendicular bisector..................................................................................106
Perpendicular to a point interior to a line segment.............................................................107
Perpendicular at end of line segment.................................................................................107
Bisecting an angle...............................................................................................................108
Inaccessible point...............................................................................................................108
Laying out right angles........................................................................................................109
Copying an existing angle...................................................................................................109
Dividing an angle................................................................................................................110
Laying out angles................................................................................................................114
Using tangents................................................................................................................114
60°..................................................................................................................................115
45°..................................................................................................................................116
30°, 15°...........................................................................................................................116
Other special angles.......................................................................................................116
Triangles.............................................................................................................................116
Inscribed circle................................................................................................................116
Circumscribed circle.......................................................................................................116
Hexagon..............................................................................................................................116
Octagon..............................................................................................................................117
Ellipse..................................................................................................................................119
Piece of string and two foci............................................................................................120
Calculated points............................................................................................................120
Using a marked stick......................................................................................................120
Approximate Ellipse with circles.....................................................................................121
Parabola..............................................................................................................................124
Regular polygons................................................................................................................124
Arcs.....................................................................................................................................125
Tapered boxes....................................................................................................................126
Useful tips................................................................................................................................127
References..............................................................................................................................127
Copyright
Copyright © 2010 Don Peterson
Permission to use, copy, modify, distribute and sell this document for any purpose is hereby
granted without fee, provided that the above copyright notice, this paragraph, and the following
paragraph appear unchanged in all copies.
The author is identified by an SHA-256 hash of the following information: the string "Don
Peterson", a number familiar to the author (hint: first Honda), and a password (Gary E.'s) are
concatenated and put into a UNIX-style file. The resulting hash is
b31c011d1fd2046af4e2eb4c9b40a82e4fc8de6f8cbbc00c0fd5bd54ad52583c.
Notation
All angle measure is assumed to be in radians unless the degree symbol ° is present or a
statement to that effect is given. mrad is milliradians and rad is microradians.
log means the base 10 logarithm; ln means the natural logarithm.
A vector is given in bold: A; the magnitude of A is A. All scalars are real unless otherwise stated.
r is the position vector of a general point in space = (x, y, z), r1 = (x1, y1, z1), etc.
A caret denotes a unit vector and it is gotten by dividing a vector by its magnitude:
v
v =
v
The usual Cartesian unit vectors are  i , j , k .
iff means "if and only if". Special formatting is used to make it stand out, as it can be easy to miss
the second f.
n denotes a normal vector; what it is normal to will be clear from the context.
An overdot denotes differentiation (Newton's notation). It will either be respect to a single variable
or a parameter. Multiple dots denote higher derivatives.
∂f
= ∂ x f.
∂x
Partial derivatives: f x =
In parametric equations, s, t, and u are parameters and s ,t ,u ∈ℝ.
 =  ,  ,  where , , and  are the three direction cosines of a direction in space.
The Levi-Civita symbol is
ijk =
{
1 if (i,j,k) is (1,2,3), (3,1,2), or (2,3,1) (even permutation)
−1 if (i,j,k) is (1,3,2), (3,2,1), or (2,1,3) (odd permutation)
0 otherwise
or
i=1
0 0
0
0 0
1
0 −1 0
∣
i
0
0
1
=2
0 1
0 0
0 0
i =3
0
1 0
−1 0 0
0
0 0
∣∣ ∣∣
∣
When the Levi-Civita symbol is used, the usual Einstein summation convention is assumed over
the three spacial indices 1, 2, and 3.
ijk klm = il  jm− i m  jl
[xyz] is a reference to reference xyz. [xyz:m] is a reference to page m in reference xyz. [xyz:m:p]
uses p to point to the page number in the PDF for documents that are accessible in PDF format.
Constants
The following two pages contain some common constants. The first page gives the numbers in a
conventional order and the second page gives them sorted by their significands. The beginning
digit frequencies look a little like Benford's "Law". For a more extensive list, see e.g. H. Robinson
and E. Potter, Mathematical Constants, Lawrence Radiation Laboratory, UCRL-20418, March 1971
(this was also published in Mathematics of Computation, Vol. 26, Jan 1972).
3.142
6.283
9.425
12.57
15.71
18.85
21.99
25.13
28.27
1.571
1.047
0.7854
0.6283
0.5236
0.4488
0.3927
0.3491
0.3183
0.6366
0.9549
1.273
1.592
1.910
2.228
2.546
2.865
0.01745
57.30
2.094
4.189
2.356
0.1592
0.1061
0.07958
9.870
31.01
97.41
1.772
0.8862
0.5908
0.4431
1.465
1.331
0.5642
1.128
0.6828
0.3989
0.4971

2
3
4
5
6
7
8
9
/ 2
/3
/ 4
/5
/6
/7
/8
/9
1/
2/
3/
4/
5 /
6/
7 /
8/
9/
/180
180 /
2 /3
4 / 3
3/4
1/2 
1/3 
1 /4 
2

3
4

  /2
  /3
 / 4
3 
4 
1/ 
2/  
3
1/ 
1/  2
log
0.3010
0.4771
0.6021
0.6990
0.7782
0.8451
0.9031
0.9542
1.145
0.6931
1.099
1.386
1.609
1.792
1.946
2.079
2.197
2.303
0.4343
2.718
0.8653
1.156
0.3679
1.649
0.6065
1.396
0.7165
7.389
0.1353
22.46
23.14
0.04321
535.5
0.001867
4.810
0.2079
0.5000
0.3333
0.6667
0.2500
0.7500
0.2000
0.4000
0.6000
0.8000
0.1667
0.8333
0.1429
log 2
log 3
log 4
log 5
log 6
log 7
log 8
log 9
ln 
ln 2
ln 3
ln 4
ln 5
ln 6
ln7
ln 8
ln 9
ln 10
log e
e
e /
/e
1/e
e
1/  e
3 e
3
1/  e
2
e
e−2
e

e
e−
e2 
−2 
e
/2
e
e−/2
1/2
1/3
2/3
1 /4
3 /4
1/5
2/5
3/5
4/5
1/6
5/6
1/7
0.2857
0.4286
0.5714
0.7143
0.8571
0.1250
0.3750
0.6250
0.8750
0.1111
0.2222
0.4444
0.5556
0.7778
0.8889
0.1000
0.3000
0.7000
0.9000
0.09091
0.1818
0.2727
0.3636
0.4545
0.5455
0.6364
0.7273
0.8182
0.9091
0.08333
0.4167
0.5833
0.9167
0.07692
0.1538
0.2308
0.3077
0.3846
0.4615
0.5385
0.6154
0.6923
0.7692
0.8462
0.9231
0.07143
0.2143
0.3571
2/7
3/7
4/7
5 /7
6/7
1 /8
3 /8
5 /8
7 /8
1 /9
2/9
4/9
5 /9
7 /9
8 /9
1 /10
3 /10
7 /10
9 /10
1 /11
2/11
3/11
4/11
5 /11
6/11
7 /11
8 /11
9 /11
10 /11
1 /12
5 /12
7 /12
11 /12
1 /13
2/13
3 /13
4/13
5 /13
6/13
7 /13
8 /13
9 /13
10 /13
11 /13
12 /13
1 /14
3 /14
5 /14
0.6429
0.7857
0.9286
0.06667
0.1333
0.2667
0.4667
0.5333
0.7333
0.8667
0.9333
0.06250
0.1875
0.3125
0.4375
0.5625
0.6875
0.8125
0.9375
0.05882
0.1176
0.1765
0.2353
0.2941
0.3529
0.4118
0.4706
0.5294
0.5882
0.6471
0.7059
0.7647
0.8235
0.8824
0.9412
0.05556
0.2778
0.3889
0.6111
0.7222
0.9444
9/14
11/14
13/14
1/15
2/15
4/ 15
7/15
8/15
11/15
13/15
14/15
1/16
3/16
5/16
7/16
9/16
11/16
13/16
15/16
1/17
2/17
3/17
4/ 17
5/17
6/17
7/17
8/17
9/17
10/17
11/17
12/17
13/17
14/17
15/17
16/17
1/18
5/18
7/18
11/18
13/18
17/18
1.047
/3
0.1061
1/3 
1.099
ln 3
0.1111
1/9
1.128
2/  
1.145
ln 
1.156
/e
0.1176
2/17
0.1250
1/8
12.57
4
1.273
4/
1.331
4 
0.1333
2/15
0.1353
e−2
1.386
ln 4
1.396
3 e
0.1429
1/7
1.465
3 
0.1538
2/13
1.571
/2
15.71
5
0.1592
1/2 
1.592
5 /
1.609
ln 5
1.649
e
0.1667
1/6
0.01745
/180
0.1765
3/17
1.772

1.792
ln 6
0.1818
2/11
−2 
0.001867 e
0.1875
3/16
18.85
6
1.910
6 /
1.946
ln 7
0.2000
1/5
0.2079
e−/2
2.079
ln 8
2.094
2/3
0.2143
3/14
2.197
ln 9
21.99
7
0.2222
2/9
2.228
7 /
e
22.46

2.303
ln 10
0.2308
3/13

23.14
e
0.2353
4/17
2.356
3 /4
0.2500
1/ 4
25.13
8
2.546
8 /
0.2667
4/15
2.718
e
0.2727
3 /11
0.2778
5 /18
28.27
9
0.2857
2/7
2.865
9 /
0.2941
5 /17
0.3000
3 /10
0.3010
log 2
0.3077
4/13
31.01
3
0.3125
5 /16
3.142

0.3183
1 /
0.3333
1 /3
0.3491
/9
0.3529
6 /17
0.3571
5 /14
0.3636
4/11
0.3679
1 /e
0.3750
3 /8
0.3846
5 /13
0.3889
7 /18
0.3927
/8
0.3989 1/  2
0.4000
2/5
0.4118
7 /17
0.4167
5 /12
4.189
4/3
0.4286
3 /7
0.04321
e−
0.4343
log e
0.4375
7 /16
0.4431
 / 4
0.4444
4/9
0.4488
/7
0.4545
5 /11
0.4615
6 /13
0.4667
7 /15
0.4706
8 /17
0.4771
log 3
4.810
0.4971
0.5000
0.5236
0.5294
0.5333
535.5
0.5385
0.5455
0.05556
0.5556
0.5625
0.5642
0.5714
57.30
0.5833
0.05882
0.5882
0.5908
0.6000
0.6021
0.6065
0.6111
0.6154
0.06250
0.6250
0.6283
6.283
0.6364
0.6366
0.6429
0.6471
0.06667
0.6667
0.6828
0.6875
0.6923
0.6931
0.6990
0.7000
0.7059
0.07143
0.7143
0.7165
0.7222
0.7273
0.7333
7.389
/2
e
log
1/2
/6
9/17
8/15
e2 
7/13
6/11
1/18
5/9
9/16
1/ 
4/7
180 /
7/12
1/17
10/17
 /3
3/5
log 4
1/  e
11/18
8/13
1/16
5/8
/5
2
7/11
2/
9/14
11/17
1/15
2/3
3
1/ 
11/16
9/13
ln 2
log 5
7/10
12/17
1/14
5/7
3
1/  e
13/18
8/11
11/15
e2
0.7500
3 /4
0.7647
13 /17
0.07692 1 /13
0.7692
10 /13
0.7778
7 /9
0.7782
log6
0.7854
/4
0.7857
11 /14
0.07958 1/4 
0.8000
4/5
0.8125
13 /16
0.8182
9 /11
0.8235
14 /17
0.08333 1 /12
0.8333
5 /6
0.8451
log7
0.8462
11 /13
0.8571
6/7
0.8653
e /
0.8667
13 /15
0.8750
7 /8
0.8824
15 /17
0.8862
 / 2
0.8889
8 /9
0.9000
9 /10
0.9031
log8
0.9091
10 /11
0.09091 1 /11
0.9167
11 /12
0.9231
12 /13
0.9286
13 /14
0.9333
14 /15
0.9375
15 /16
0.9412
16 /17
9.425
3
0.9444
17 /18
0.9542
log9
0.9549
3 /
97.41
4
9.870
2
0.1000
1 /10
Series, approximations, and formulas
An arithmetical progression of n terms is [gr:1]
a[ ar ][a2r ][a3r ][an−1 r ]
n −1
n
n
= ∑ ak r  = [2an−1r ] = aL where L is the last term
2
2
k =0
A geometric progression is [gr:1]
aaqaq 2 aq 3aq n−1
n
aq n−1
= ∑ aq k−1 =
q−1
k =1
The binomial theorem and its generalization lead to useful approximations. The theorem is [gr:21]
1x n = 1nx
n n−1 2 n n−1n−2 3
n!
x 
x 
x k 
2!
3!
n−k ! k !
If n is a positive integer, the series has a finite number of terms; otherwise there are an infinite
number of terms. If n is not an integer, the series converges absolutely for ∣x∣  1 and diverges for
∣x∣  1. For x = 1, the series converges for n  −1 and diverges for n ≤ −1 and the series
converges absolutely for n  0. For x =−1, it converges absolutely for n  0 and diverges for
n  0.
For integer n > 0 [gr:21]
n
x y  = ∑
n
k= 0

n n− k k
x y
k
where the binomial coefficient is

n!
n
=
k
n−k  ! k !
Some special forms are [gr:21]
∞
1
2
3
4
k −1
= 1  x  x  x  x = ∑ x
1−x
k =1
∞
1
2
3
4
= 1 − x  x − x  x −  = ∑ −1 k−1 x k− 1
1x
k =1
∞
1
2
3
= 1 ∓ 2x  3x ∓ 4x  = ∑ −1k −1 k x k −1
2
1±x 
k =1
∞
x
k
2 = ∑ k x where x  1
1−x 
k =1
1
1⋅1
1⋅1⋅3
1⋅1⋅3⋅5
 1±x = 1 ± 2 x − 2⋅4 x 2 ± 2⋅4⋅6 x 3 − 2⋅4⋅6⋅8 x 4 ± 
1
1
1 3
5 4
7 5
21 6
= 1 ± x − x2 ±
x −
x ±
x −
x ±
2
8
16
128
256
1024
1
1
1⋅3 2 1⋅3⋅5 3 1⋅3⋅5⋅7 4
=1 ∓ x 
x ∓
x 
x ∓
2
2⋅4
2⋅4⋅6
2⋅4⋅6⋅8
 1±x
1
3
5 3
35 4
63 5
231 6
= 1 ∓ x  x2 ∓
x 
x ∓
x 
x ∓
2
8
16
128
256
1024
These expansions lead to commonly-used approximations when x is small, such as
1
≈1∓x
1±x
1
≈ 1 ∓ 2x
2
1±x 
 1±x = 1 ± 1 x
2
1
1
=1∓ x
2
 1±x
These generalize to (a, b, , and  can be complex)
1±a 1±b ≈ 1±a ±b where ∣∣≪1 and ∣∣≪1
Stirling's formula
lim
n ∞
n!
=  2 = 2.5066
n e−n  n
n
If x and y are nearly equal, another occasionally-useful approximation is  xy ≈ x y /2.
Sums of integers [gr:1]
n
∑kp
p
k =0
1
1
n n1
2
2
1
n n1 2n1
6
[
3
1
nn1
2
2
]
4
1
2
n n12n13n 3n−1
30
5
1 2
2
2
n n1 2n 2n−1
12
n
∑ 2k−1p
p
k =1
1
n2
2
1
2
n 4n −1
3
3
n 2 2n2−1
Other series
∞
a
∑ aqk = 1−q
for ∣q∣1
k=0
∞
a
rq

∑ akr  q k = 1−q
1−q 2
k=0
∞
p
∑ kq k = 1−p
2
k=0
for ∣q∣1
for ∣q∣1
x2 x3
 
2! 3!
x lna2 x ln a3
a x = 1x ln a


2!
3!
x3 x5 x7
x3 x5
x7
x9
sin x = x −  −  = x − 
−

−
3! 5! 7!
6 120 5040 362880
x2 x4 x6
x2 x4
x6
x8
x 10
cos x = 1−  −  = 1−  −

−

2! 4 ! 6 !
2 24 720 40320 3628800
1
2 5 17 7
62 9
tan x = x  x 3
x
x 
x  for ∣x∣  1
3
15
315
2835
 1 1
1
= −  3 − 5  for ∣x∣  1
2 x 3x 5x
x2 x 3 x 4
ln1±x  = ±x − ± − ±
2 3 4
x
e = 1x 
Newton's method applied to calculating roots: if x n = N and if xk is an approximation of N1/n, then
the sequence [as:18]
x k1 =
[
1 N
n−1x k
n x n−1
k
]
will converge quadratically (i.e., about double the number of significant figures every iteration) to x.
For square roots, this gives (sometimes called the Babylonian method)
x k1 =
[
]
[
]
1 N
x k
2 xk
For cube roots
x k1 =
1 N
2 x k
3 x 2k
Trigonometry
Many trigonometric identities can be derived using Euler's identity (see [feynman])
e i  = cos i sin 
Example: the addition formulas for the sine and cosine can be calculated from e i x e iy:
e ix e iy = cos x i sin x cos y i sin y 
e i  x y  = cos x cos y −sin x sin y i  sin x cos y cos x sin y 
cosx y i sin xy  = cos x cos y −sin x sin y i sinx cos y cos x sin y 
and the addition formulas can be read off by equating real and imaginary parts. Then, knowing the
sine is odd and the cosine is even, substitute -y for y and get the difference formulas.
The trig functions on the unit circle are (you can get the script that produces this drawing from
http://code.google.com/p/hobbyutil/; download the TrigUnitCircle.zip file):
Figure 1
Some seldom-used trig functions are:
exsec 
vers
covers 
hav 
In the figure
sec −1
1−cos (versed sine)
1−sin
1
vers  (haversine)
2
rs
ec
θ
a
b
r
y
θ
x
y
1
=
= cos
r
csc 
x
1
cos  = =
= sin
r
sec 
sin  =
tan=

e i −e−i 
− =
2
2
i
−i 

e e
− =
2
2
 
 

i
−i 
y
sin
1
e −e
=
=
= −i i  −i 
x
cos cot 
e e
a =  r b −r
2

2
b = r tan  = r
y
= r asin
x
The distance r + a is immediately seen to be equal to r sec  because of the Pythagorean theorem
with the right triangle with the vertical b = r tan  leg and the other leg of r.
Relations between the circular functions
Relations between the circular functions (or inverse circular functions) for 0 ≤ x ≤
sin x = a
cos x = a
sin x
a
 1−a2
cos x
 1−a2
a
tan x
a
 1−a 2
 1−a 2
a
tan x = a
a
 1a 2
1
 1a 2
a

:
2
Magnitude and sign
If x is in quadrant
2
3
+1 to +0
-0 to -1
-0 to -1
-1 to -0
-∞ to -0
+0 to +∞
-0 to -∞
+∞ to +0
-∞ to -1
-1 to -∞
+1 to +∞
-∞ to -1
1
+0 to +1
+1 to 0
+0 to +∞
+∞ to +0
+1 to +∞
+∞ to +1
sin x
cos x
tan x
cot x
sec x
csc x
4
-1 to -0
+0 to +1
-∞ to -0
-0 to -∞
+∞ to +1
-1 to -∞
Angles in other quadrants
For an angle x in the first quadrant, the following table gives the functions in other quadrants:
sin x
cos x
tan x
cot x
sec x
csc x
-x
−sin x
cos x
−tan x
−cot x
sec x
−csc x
/2 ± x
cos x
∓sin x
∓cot x
∓tan x
∓csc x
sec x
±x
∓sin x
−cos x
±tan x
±cot x
−sec x
∓csc x
3/2 ± x
-cos x
±sin x
∓cot x
∓tan x
±csc x
−sec x
n ± x
+- sin x
cos x
±tan x
±cot x
sec x
±csc x
Small angle approximations
For small , the Taylor series give
sin  ≈ 
tan ≈ 
cos  ≈ 1−
x2
2
Ordering relations
From [perry:2-21]
sin     tan 
sin 
cos  
1

 1−2  sin   1
2
  1−  sin   

cos  
1
tan
 
2
 sin  
2

  tan  
 1−2
sin
lim
=1

x ∞
sin
lim
=1
x ∞ tan 
 1−
Values for some special angles
The rows with the light yellow background were derived by Herman Robinson (1912-1986).
Rad
0

24
Deg
0°
7.5°
sin x = a
0
cos x = a
1
tan x = a
0
 8−2  2−2  6  82  22  6  6− 3 2−2
4
4

16
11.25°
 2−  2 2
 2  2 2

12
15°
 2   3−1
 2  31

10
18°
 5−1
 102  5

8
22.5°

6
30°

5
36°

4
45°

3
60°
5
12

2
2
4
4
2
4
4
 2−  2
 2  2
2
1
2
2
3
2
 10−2  5
 51
 42  2− 2−1
2−  3

1−
2
5
 2−1
3
3
 5−2  5
4
2
2
3
2
4
2
2
1
2
75°
 2  31
 2   3−1
2 3
90°
1
0
∞
4
4
1
3
Pythagorean relationships
sin2 x cos2 x = 1
1tan 2 x = sec 2 x
Sums and products
sinx ±y  = sin x cos y ±cos x sin y
cosx ±y  = cos x cos y ∓sin x sin y
tan x ±tan y
tanx ±y  =
1∓tan x tan y
sinx y sinx −y  = sin2 x −sin2 y = cos2 y −cos2 x
cosx y cos x −y  = cos2 x −cos 2 y = cos2 y −sin2 x
1
1
sin x ±sin y = 2sin  x ±y cos x ∓y 
2
2
1
1
cos x cos y = 2cos x y cos x −y 
2
2
1
1
cos x −cos y =−2sin x y sin x −y 
2
2
sinx ±y 
tan x ±tan y =
cos x cos y
1cot 2 x = csc 2 x
1
1
cos x−y − cos x y 
2
2
1
1
cos x cos y = cosx −y  cos xy 
2
2
1
1
sin x cos y = sin xy  sinx −y 
2
2
sin x sin y =
Multiple angles
sin 2 x = 2 sin x cos x =
2 tan x
2
1tan x
cos 2 x = cos2 x −sin2 x = 2 cos2 x −1 = 1−2 sin2 x =
2 tan x
1−tan 2 x
sin 3 x = 3 sin x −4 sin3 x
cos 3 x = 4 cos3 x −3 cos x
2
sin 4 x = 4 sinx cos x −8 sin x cos x
4
2
cos 4 x = 8 cos x−8 cos x1
sin n x = 2 sinn−1 x cos x −sinn−2x
cos n x = 2cosn−1x cos x −cosn−2x
tann−1 x tan x
tan n x =
1−tan n−1x tan x
tan 2 x =
Half angles



x
=±
2
x
cos = ±
2
x
tan = ±
2
sin
1−cos x
2
1cos x
2
1−cos x 1−cos x
sin x
=
=
1cos x
sin x
1cos x
Powers
1
1−cos 2 x 
2
1
sin3 x = 3 sin x −sin3x
4
1
4
sin x = cos 4x−4 cos2x3
8
1
2
cos x = 1cos 2x
2
1
cos3 x = cos 3x3 cos x 
4
1
cos4 x = cos 4x4 cos2x3
8
2
sin x =
1−tan2 x
2
1tan x
Inverse functions
Principal values
Domain
[
−1 ≤ x ≤ 1
sin −1 x
Range
− 
,
2 2
−1
−1 ≤ x ≤ 1
[ 0 , ]
−1
tan x
ℝ
ℝ
cot −1 x
ℝ
 0 , 
cos x
[ 0 ,2 
[− ,2 
0 ,2 ]
− ,2 ]
x ≥1
sec−1 x
x ≤−1
x ≥1
−1
csc x
]
x ≤−1
Relations
These are for the principal values.

2

−1
−1
tan x  cot x =
2
−1
−1

sec x  csc x =
2
sin−1 x  cos−1 x =
2
x
1−x
1
1
= cot −1 
= sec−1
= csc−1
2
2
x
x
 1−x
 1−x
2
x
1
−1
−1
2
−1  1−x
−1
−1 1
−1
cos x = sin  1−x = tan
= cot
= sec
= csc
2
x
x
 1−x
 1−x 2
sin−1 x = cos−1  1−x 2 = tan−1
x
1
1
1x
= cos−1
= cot −1 = sec−1  1x 2 = csc−1 
2
2
x
x
 1x
 1x
1
cot −1 x =tan−1
x
1
sec−1 x =cos−1
x
−1
−1 1
csc x =sin
x
tan−1 x = sin−1
2
2
sin x ± sin y = sin x  1−y ± y 1−x 
cos−1 x ± cos−1 y = cos−1 xy ∓  1−x 2  1−y 2 
x ±y
tan−1 x ± tan−1 y = tan−1
1 ∓ xy
−1
−1
−1
2
cossin x  = sincos x  =  1−x
tansec−1 x  = cotcsc−1 x  =  x 2−1
sec tan−1 x  = csc cot −1 x  =  x 21
−1
−1
2
 
x

x
−1
−1
−1
−1
= −cos x =−sin −x  = tan
= −cot
2
2
2
 1−x 2
 1−x 2
x
x
−1

−1

−1
−1
−1

−1
cos x = sin −x  = −sin x = −cos −x  = cot
= −tan
2
2
2
 1−x 2
 1−x 2
 
x

x
tan−1 x = cot −1 −x − = −cot−1 x = −tan−1 −x  = sin−1
== −cos−1 2
2
2
2
2
 x 1
 x 1
−1
−1
sin x = cos −x −
Triangles
B
a
c
r
hb
A
C
b
R
We'll use the notation
a, b, c = triangle's sides
A, B, C = triangle's corresponding angles
K = area
r = radius of inscribed circle (incircle)
R = radius of circumscribed circle (circumcircle)
hb = altitude that intersections side b (and analogous for the other sides)
* denotes a formula that has other forms which are gotten by cyclically permuting the variables (a,
b, c) and (A, B, C).
s=
1
abc 
2
Q = s −as−bs −c
ABC = 
a
b
c
=
=
sin A sinB
sinC
1
tan  AB
ab
2
=
*
a−b
1
tan  A−B
2
a 2 = b 2c 2−2 bc cos A *
a = b cosCc cosB *
cos A =
b 2c 2−a 2
*
2 bc
sin A =
2
 sQ *
bc



sin
s−b s −c 
A
=
*
2
bc
cos
s s−a
A
=
*
2
bc
tan
s−bs −c 
A
=
*
2
s  s−a
h b = c sin A * = a sinC * =
r=

R=
2
 sQ *
b
Q
A
K
* = s−atan
*=
s
2
s
ac
a
abc
*=
*=
2 hb
2 sin A
4K
1
1
a 2 sinB sinC
b hb * = a b sinC * =
* =  sQ = r s
2
2
2 sin A
A
B
C
= 2 R2 sin A sinB sinC = s 2 tan tan tan
2
2
2
K=
sin A  sinB  sinC = 4cos
A
B
C
cos cos
2
2
2
tan A  tan B  tanC = tan A tan B tan C
ab  c , bc  a , ac  b, ∣a−b∣  c , ∣b−c∣  a , ∣a−c∣  b
The center of the circumscribed circle is at the intersection of the perpendicular bisectors of
the sides. The center of the inscribed circle is at the intersection of the bisectors of the internal
angles.
2
Distance of center of circumscribed circle to center of inscribed circle =  R −Rr
The lines from the angles' vertex to the midpoints of the sides intersect at the centroid. The
distance of the centroid from one side is equal to one-third of the side's associated altitude. The
1
2
2
2
length of the line from A to the midpoint of a is  2 b c −a with analogous formulas for the
2
other two sides/angles.
The length of the angle bisector for angle A is
the other two sides.
1
 bc [bc 2−a 2 ] with analogous formulas for
bc
The altitudes are in the ratios h a : hb : h c =
1 1 1
: : .
a b c
Solution of right triangles
Given any two sides or one side and an acute angle A, find the remaining parts of the triangle.
B
c
a
A
b
sin A =
a
b
a

, cos A = , tan A = , B= −A
c
c
b
2
2
2
2
2
a =  c −b =  c bc−b = c sin A = b tan A
b =  c −a =  c ac −a = c cos A =
c=
a
tan A
a
b
2
2
=
=  a b
sin A cos A
Area =
1
a2
b 2 tan A c 2 sin2 A
ab =
=
=
2
2 tan A
2
4
Some commonly-used right triangles are:
π/2
45°
√2
1
π/2
45°
π/6
30°
2
√3
π/3
60°
1
1
The Pythagorean triples that are less than 300 are:
5
1.55π
36.87° 4
1.078π
53.13°
3
a
3
5
7
8
9
11
12
13
15
16
17
19
20
20
21
23
24
28
28
32
36
39
44
b
4
12
24
15
40
60
35
84
112
63
144
180
21
99
220
264
143
45
195
255
77
80
117
c
5
13
25
17
41
61
37
85
113
65
145
181
29
101
221
265
145
53
197
257
85
89
125
a
48
51
52
60
60
65
68
69
84
85
88
95
96
104
105
115
119
120
133
140
160
161
b
55
140
165
91
221
72
285
260
187
132
105
168
247
153
208
252
120
209
156
171
231
240
c
73
149
173
109
229
97
293
269
205
157
137
193
265
185
233
277
169
241
205
221
281
289
A Pythagorean triple is such that a 2b 2 = c 2. This list only contains triples that are not integer
multiples of other triples (i.e., a and b are relatively prime).
Solution of oblique triangles
B
c
a
C
A
b
ASA
Given angles A and B and side c:
C = −AB )
a=
b=
c sin A
sinC
c sin B
sinC
SSA
Given sides a and c and angle A:
sinC =
c sin A
a
B = −AC 
b=
a sinB
sin A
C may have two values: C 1 


and C 2 = −C 1  . If AC 2   , using only C1.
2
2
SAS
Given any two sides b and c and their included angle A. Use any one of the following sets of
formulas:
(1)
1
A
BC = −
2
2
1
b−c
1
tan  B−C =
tan BC 
2
bc
2
1
1
B = BC B−C
2
2
1
1
C = BC− B−C
2
2
b sin A
a=
sinB
(2)
a =  b c −2bc cos A
b sin A
sin B =
a
C = −AB 
(3)
c sin A
b−c cos A
B = −AC 
c sin A
a=
sinC
2
2
tanC =
SSS
Given the three sides a, b, and c. Use either of the following sets of formulas.
(1)
1
abc 
2
 s−a s −bs−c 
r=
s
A
r
tan =
2
s−a
B
r
tan =
2
s−b
C
r
tan =
2
s−c
s=

(2)
b 2c 2−a 2
2bc
c 2a 2−b 2
cos B =
2ca
C = −AB 
cos A =
Mensuration
In the following
n = number of sides
r = radius of inscribed circle
R = radius of circumscribed circle
A = area
K = area (used on e.g. triangles where A is one of the angles)
p = perimeter
V = volume
h = height
Two-dimensional figures
Rectangle
a
θ
b
2
2
2
2
A = a b = a  d −a = b  d −b =
1 2
d sin
2
d = length of diagonal =  a2b2
p = 2ab 
2
2
a =  d −b
b =  d 2−a2
Right triangle
c
b
a
1
ab
2
p = abc
c 2 = a 2b 2
A=
Isosceles triangle
a
θ
a
h
b
1
bh
2
p = 2ab
A=
b = a  21−cos  = a sin
h = a cos

2

2
General triangle
We use K for the area because A is used for one of the angles.
C
b
a
h
m
t
B
A
midpoint
c
1
1
c 2 sin Asin B
abc
c h = ab sinC =
= rs =
=  s s −as−bs−c 
2
2
2 sinC
4R
A
B
C
= r 2 cot cot cot = 2 R 2 sin A sinB sinC
2
2
2
1
= [ x 1 y 2−x 2 y 2 x 2 y 3−x 3 y 2 x 3 y 1−x 1 y 3 ]
2
where x 1 , y 1 ,x 2 , y 2 ,∧x 3 , y 3  are coordinates of the vertices
1
p
s = abc  =
2
2
A
B
C
ab sinC
C
abc
r = c sin sin sec =
= s −c tan =
2
2
2
2s
2
4Rs
 s−as −bs−c  K
A
B
C
=
= = 4R sin sin sin
s
s
2
2
2
a
b
c
abc
abc
R=
=
=
=
=
2 sin A 2 sinB
2 sinC
 s s−as−b s −c  4K
2K
2
h = a sinB = b sin A =
=  s s−a s−bs−c 
c
c
K = area =

t = length of bisector of angle C =

[
2ab
C
c2
cos = ab 1−
ab
2
ab2
m = length of line connecting to c's midpoint =
a
b
c
=
=
(Sine law)
sin A sinB
sinC
c 2 = a2b2−2ab cosC (Cosine law)

]
1 2 2 1 2
a b − c
2
4
The median lines m that join each vertex with the midpoint of the side opposite meet in the centroid
of the triangle; the centroid trisects each median.
The altitudes meet at a point called the orthocenter. It lies within the triangle iff the triangle is
acute (i.e., doesn't have an angle larger than a right angle). If the triangle is a right triangle, the
orthocenter is the vertex of the right angle. If the vertexes of the triangle are the points A, B, C and
the orthocenter is point H, then we have the relation
a 2b 2c 2AH 2BH 2CH 2 = 12 R 2
The perpendicular bisectors of the sides meet at the center of the circumscribed circle with radius
R.
The angle bisectors meet at the center of the inscribed circle with radius r.
If c is the largest side of the triangle, then C is the largest angle and ∣a−b∣  c  ab.
Parallelogram
b
a
φ
a
h
θ
b
1
d d sin
2 1 2
where d 1 and d 2 are the diagonals and  is the angle between them
p = 2ab 
A = b h = a b sin =
Trapezoid
a
h
b
A=
1
hab
2
Note the two vertical sides do not have to intersect the base at the same angle.
Trapezium
H
h
b
a
A=
c
H hab hc H
2
Quadrilateral
For a quadrilateral inscribed in a circle
b
a
θ
d2
c
d1
d
1
1
d 1 d 2 sin =  s −as−bs−c s−d  = acbd sin
2
2
where d 1 and d 2 are the diagonals and  is the angle between them
p = abc d
p
s=
2
A=
For a general quadrilateral
b
a
θ
d2
c
d1
d
A=
1
d d sin
2 1 2
where d 1 and d 2 are the diagonals and  is the angle between them. If m is the distance between
the midpoints of d 1 and d 2, then
a 2b 2c 2d 2 = d 12d 22 4 m 2
General polygon
1
2
3
4
5
Divide the polygon into triangles and sum their areas; for example:
A = A1A2A3 A 4A5
Hexagon (regular)
D
R
s=R
r
d
r = radius of inscribed circle =
R
 3 R = 0.8660R
2
2
r = 1.155r
3
 3 D = 0.8660D
d = distance across flats = 2 r =
2
2
D = distance across points = 2 R =
d = 1.155 d
3
1
3
3 3
A = 6 R r =  d 2 = 0.8660d 2 =  D 2 = 0.6459D 2
2
2
8
3

3
= 2  3 r 2 = 3.454 r 2 =
R 2 = 2.598R 2
2
6
p = 6s = 6R = 3 D =
d = 3.464d
3
R = radius of circumscribed circle=
 
Octagon (regular)
D
R
r
R
d
s
r = radius of inscribed circle = R cos
 8  =  22  2 R = 0.9239R
2r
= 1.0324 r
 2  2
d = distance across flats = 2 r = D  2  2 = 1.8478D
¿
4d
D = distance across points = 2 R =
= 2.1648d
 2  2 2
1
D
A = 8 R r = 2d 2   2−1 = 0.8284 d 2 =
= 0.7071D 2 = 2 s 2   21
2
2
s =d   2−1 = 0.4142d
p = 8s
R = radius of circumscribed circle=
 
r
Polygon (regular)
R
θ
β
s
n = number of sides
2
=
n
n−2
=

n


s = 2r tan = 2 R sin
n
n
p = ns
1

 1
2
A = ns 2 cot = nr 2 tan = nR 2 sin
4
n
n
2
n
s


r = cot = R cos
2
n
n
s


R = csc = r sec
2
n
n
The following table gives various polygon characteristics; multiply by the column's denominator to
get the value for a polygon of arbitrary size. The symbols are defined above. [marks:1-39]
n
3
4
5
6
7
8
9
10
12
15
16
20
24
32
48
64
θ
2.094
1.571
1.257
1.047
.8976
.7854
.6981
.6283
.5236
.4189
.3927
.3142
.2618
.1963
.1309
.09817
θ, ° area/s2 area/R2 area/r2
120
.4330
1.299
5.196
90
1.000
2.000
4.000
72
1.720
2.378
3.633
60
2.598
2.598
3.464
51.43 3.634
2.736
3.371
45
4.828
2.828
3.314
40
6.182
2.893
3.276
36
7.694
2.939
3.249
30
11.20
3.000
3.215
24
17.64
3.051
3.188
22.5
20.11
3.061
3.183
18
31.57
3.090
3.168
15
45.57
3.106
3.160
11.25 81.23
3.121
3.152
7.50
183.1
3.133
3.146
5.625 325.7
3.137
3.144
R/s
.5774
.7071
.8507
1.000
1.152
1.307
1.462
1.618
1.932
2.405
2.563
3.196
3.831
5.101
7.645
10.19
R/r
2.000
1.414
1.236
1.155
1.110
1.082
1.064
1.051
1.035
1.022
1.020
1.012
1.009
1.005
1.002
1.001
s/R
1.732
1.414
1.176
1.000
.868
.765
.684
.618
.518
.416
.390
.313
.261
.196
.131
.098
s/r
3.464
2.000
1.453
1.155
.9631
.8284
.7279
.6498
.5359
.4251
.3978
.3168
.2633
.1970
.1311
.0983
r/R
.5000
.7071
.8090
.8660
.9010
.9239
.9397
.9511
.9659
.9781
.9808
.9877
.9914
.9952
.9979
.9988
r/s
.2887
.5000
.6882
.8660
1.038
1.207
1.374
1.539
1.866
2.352
2.514
3.157
3.798
5.077
7.629
10.18
Circle
r
d
C = circumference = 2 r =d = 2   A =
A = r 2 =
2A
r
1
C2
1
d 2 =
= Cr
4
4 2
Circular sector
b
C
θ
x
r
A
s
b=r 
d =2r
 
2

r  br
1
=
=
= d2
2
2
2
8

s = 2r sin
2
Centroid is distance x up from center A on symmetry axis
2 rs 4 r

x=
=
sin
3 b 3
2
A = r 2
Circular segment
b
m
h
C
x
θ
r
d
G
s
b=r
s 24h2
r=
8h
 1
 1
= s cot =  4r 2−s 2
2 2
2 2
1
h = r −d = r −  4r 2−s 2 for h  r
2

= r 1−cos
2


s = 2r sin = 2r −htan = 2  2hr−h 2 = r  21−cos
2
2
m =  2rh
1
b
s
s
A = r b − r sin
= r 2 sin−1 −  4r 2−s 2
2
r
2r 4
1
1 2
A = [ br − s r −h] = r  − sin
2
2
r −h
2
−1 d
2
= r cos
− d  r −d 2 = r 2 cos−1
− r −h  2rh−h2
r
r
d = r −h = r cos




The following approximation is accurate within 1 part in 1000 for angles 

= 16 °
11
2
hs (i.e., the area of the segment is about 2/3 of the enclosing rectangle's area)
3
The centroid of the segment is a distance x up from center G on symmetry axis:
s3
x=
12 A
A≈
For measurements on a segment where the center of the arc may not be accessible, see Arcs.
For arcs where s is small, Huygen's approximation is [marks:2-17]
b≈
1
8m−s
3
Given two of the variables r, s, b, , and h, the following table [mh04:70] gives formulas for the
other variables:
Given
, r
, s
, h
, b
r, s
Formulas
s = 2r sin
r=
r=
h
s=

1−cos
2
r=
−1
 = 2 sin
s=
 
r, b
=
h
r
b
r
−1
 = 4 tan
2
r=
Iterate the following equation for :
s 24h2
8h
2 sin

2
1−cos

2

b 1−cos
h=
2
s = 2  h 2r−h
b
2r
s
h
b=

2
4r −s
2
s = 2r sin
2h
s
b=r
b=

h=r −
s
2r

2h

tan
4
2b sin
b


2
s

tan
2
4
h=

2 sin
2
 = 2 cos−1 1−
s, b
h = r 1−cos
s
r, h
s, h


2

2


 
 


b = 2r sin−1
s
2r
b = 2r cos−1 1−
h = r 1−cos
b=
h
r
b
2r
s 24h 2
−1 2h
tan
2h
s
2b

=
 Then use the other
s
sin
2
formulas for the remaining terms.
h, b
Iterate the following equation for :
b

=
h 1−cos  Then use the other
2
formulas for the remaining terms.
For the approximate area of the segment being two-thirds of the enclosing rectangle's area, an
informal Monte Carlo study gave the following histogram for angles uniformly distributed over the
indicated range:
For other angle ranges, the shape of the histogram was substantially the same. The maximum %
deviation from the correct value for the segment's area depends on the maximum allowed :
a
Maximum %
deviation for
0 <  < /a
/a in °
1
15
180
2
3.2
90
4
0.76
45
8
0.18
22
16
0.043
11
32
0.012
5.5
In other words, for angles less than /8, you can expect the approximation to be within 0.18% of
the true value with a good chance of it being substantially better, as the median is somewhat less
than half of the given maximum value.
Thus, this is a good approximation for small angles. For hand calculations, a rule of thumb is that
the approximation's error is less than 1 part in 1000 for angles less than /11 = 16 °.
Annulus
R
r
d
A = R 2−r 2  = Rr R−r  =


D
 2
D −d 2 
4


A 2
4A
r =
d 2


A
4A
r = R 2− = D 2−


R=
A ring is where one circle completely contains another; they don't need to be concentric. It has the
same formula for its area as an annulus.
Annular sector
s1
s2
θ
h
d
D
R
r
h = R−r
1
1
Rr
1
2
2
2
2
A = R −r  = D −d  =  h
= h s 1s 2
2
8
2
2
s1 = R 
s2 = r 
Spandrel
r
c
r
A=r
2
1− 4  = 0.2146r = 12 c 1− 4  = 0.1073c
2
2
2
c = r 2
Ellipse
P1(x1, y1)
B
P2(x2, y2)
A
a
2b
O
b
P3(x3, y3)
2a
Note: b ≤ a
A = a b

e = eccentricity = 1 −
2

b
a

x
x
1
ab
x 1 y 2 − x 2 y 1  
sin−1 2 − sin−1 1
2
2
a
a
−1 x 2
Area of elliptical segment P 2 P 3 A = ab cos
− x2 y2
a
−1 x 2
Area of elliptical sector O P 2 AP 3 = ab cos
a
x
ab
2
−1 x 1
Area of elliptical sector O P 2 B P 1 =
sin
− sin
2
a
a
Area of elliptical segment P 1 B P 2 =



The perimeter of the ellipse is expressed in terms of the complete elliptic function of the second
kind; see the Ellipse section. Ramanujan's 1914 approximation is
[
p ≈  ab 1

10  4−
[hudson:17:26] gives the Gauss-Kummer series
]
, where =3
2
 
a−b
ab
[
x2 x4
x6
x8
x 10
 



4 64 256 16384 65536
x2 x4 x6
x8
x 10
= ab 1 2  2  2 


2 8 16 1282 2562
1 1
= F 21 − ,− ; 1 ; x 2
2 2
2
∞
1/ 2 2n
=∑
x
n
n=0
a−b
where x =
ab
p = ab 1
[
]
]
 
For a typical non-flat ellipse where, say, b = 0.7a, x is 0.176 and the 6th power term is about 1e-7, so
the series will converge quickly for practical problems with ellipses that are only slightly flattened. For
a quite flattened ellipse where b = 0.1a, the 6th power term is around 1e-3, so it will take a few more
terms to converge.
y
x
2b
y
2a
Area of shaded segment is
−1
A = xyab sin
x
a
Parabola
C
y
B
x
y
A
The function is y = a  x, so the area is the integral from 0 to x or
A=
4
xy
3
The length of the arc ABC is
 4x2 y 2
2

2
y
2x 4x y
ln
2x
y2
2

Cycloid
L
r
d
2
A = 3 r =
3
2
d
4
L = 8r = 4d
Approximate area
Let y 0 , y 1 ,, y n be equidistant ordinates separated by distance h. Then the following approximate
rules for the area enclosed by the boundary defined by these points are:
Trapezoidal rule:
A=h
[
1
y y n y 1y 2y n
2 0
]
Simpson's rule (n must be even)
A=h
[
1
y y n 4 y 1y 3y n−1 2y 2y 4y n− 2
3 0
]
[marks:2-23] Simpson's rule can also be used to compute volumes if the ordinates y i are
interpreted as the areas of plane sections of the volume at a constant distance h apart.
h
Three-dimensional figures
Symbols used are
A = surface area
V = volume
Cube
d
D
a
a
a
A = 6a 2 = 3d2 = 2D2
d3
D3
3
V =a =
=
2 2 3 3
A
d
D
3
a = V = =
=
6
2

3
d = diagonal of cube face = a  2
3d2
D = diagonal of cube =
= a 3
2

Rectangular prism
a
D
c
b
A = 2abacbc 
V = abc
2
2
2
D = diagonal =  a b c
Cylinder
A
h
A generalized cylinder can have ends of any planar surface with area A. The height h must be
measured perpendicular to the planar surfaces.
V = Ah
If the perimeter of area A is p, then the surface area As of the cylinder is
As = ph
Hollow circular cylinder
d
D
t
h
In terms of diameters:
1
t = D−d 
2
d = D−2t
D = d 2t

V = D 2−d 2
4
= t d t 
= t D−t 
In terms of radii r = d / 2 and R = D/ 2
t =R−r
r = R −t
R = r t
2
2
V = R −r 
= t 2rt 
= t 2R−t 
Truncated right circular cylinder
h2
h
h1
r
V = r 2 h = Bh
1
h h 
2 1 2
A = lateral area = ph
B = area of base =  r 2
p = perimeter of base = 2 r
h = mean height =
Pyramid or cone
s
s
h
1
Ah
3
A = area of base
h = altitude
s = slant height
p = perimeter of base
V=
The lateral area of the regular figure is
1
ps.
2
Tetrahedron
P4
P2
P1
P3
If a tetrahedron's four corners are Pi = (xi, yi, zi), i = 1, 2, 3, 4, then the volume is [salmon:22:47]
[wilson:114:141]
∣
x1
1 x2
V=
6 x3
x4
y1
y2
y3
y4
z1
z2
z3
z4
∣
1
1
1
1
Frustum of pyramid or cone
A2
s
h
s
A1
h
 A A2  A1 A2
3 1
A1 = area of lower base
A2 = area of upper base
h = altitude
s = slant height
p = perimeter of base
V=
1
 p p 2 s. Note the bases can be moved around
2 1
in the planes containing them and the figure's volume doesn't change.
The lateral surface area of the regular figure is
A right frustum1 made from two similar regular polygons of n sides has a lateral area of
(http://en.wikipedia.org/wiki/Frustum)
A=
[
n 2 2

2
2 2
2
2
2
a 1a 2 cot  a 1−a 2 sec 4h a1a 2

4
n
2

]
where a 1 and a2 are the sides of the bases.
Regular polyhedra
[perry:2-7] and wikipedia Volume and surface area of regular polyhedra with edge s:
Surface
Volume = a s 3
Surface area = a s 2
a
a
Name
4 equilateral triangles
6 squares
8 equilateral triangles
12 pentagons
Tetrahedron
 2 /12 = 0.1179
 3 = 1.7321
Hexahedron (cube)
1
6
Octahedron
2
/3
=
0.4714

2  3 = 3.4641
Dodecahedron
157  5/ 4 = 7.6631 3  2510  5 = 20.6458
20 equilateral triangles Icosahedron
5 3 5/12 = 2.1817
5  3 = 8.6603
Obelisk
The obelisk is the frustum of a rectangular pyramid:
b1
a1
h
b
a
V=
h
h
2aa1 b2a1ab1 ] = [ abaa1 bb1 a 1 b 1 ]
[
6
6
Parallelepiped
Ref. http://en.wikipedia.org/wiki/Parallelepiped
c
γ
h
β
α
b
a
V = abc  12coscoscos −cos −cos −cos 
2
2
2
If a, b, and c are vectors lying along the length of the corresponding sides, then the volume can be
gotten from the scalar triple product
1 The word is Latin for "piece" or "crumb". Note it's often misspelled "frustrum".
V = ∣a⋅b×c ∣ = h∣a×b∣
∣ [ ]∣
a1 a 2 a 3
= det b 1 b 2 b 3
c1 c2 c3
where the indexed quantities are the Cartesian components of the vectors.
The area of the base is
Abase = ∣a×b∣ = a b sin
and similar formulas hold for the other sides. Thus, the surface area is
A = 2 [ ∣a×b∣∣a×c∣∣b×c∣ ]
Prismoidal formula
A2
h
A1
Bases are parallel planes, lateral faces are triangles or trapezoids. [hudson:19:28]
h
 A A24 Am 
6 1
A1 = area of lower base
A2 = area of upper base
Am = area of midsection
h = altitude
V=
[marks:2.22] This formula is exact for any solid lying between two parallel planes and such that the
area of a section at a distance x from one of these planes is expressible as a polynomial not higher
than the third degree in x. It is approximately true for many other solids.
Wedge
c
h
b
a
V=
bh
2ac 
6
Sphere
r
d
4 3
 3
3
3
r = 4.1888r = d = 0.5236d
3
6
A = 4 r 2 = 12.57 r 2 = d 2 = 3.1416 d 2
3 6V
3
A
d=
= 1.2407  V =
= 0.5642  A


V=

The volume of the sphere is

2
of a circumscribed cylinder whose height is the sphere's diameter.
3
Spherical sector
c
θ
h
r
d
2 2
1
2
2
2
r h = 2.0944 r h = d h = 0.5236 d h
3
6
 4 3 2 3
1
Volume of wedge =
r = r =
 d3
2 3
3
12
c
A = r 2h = area of conical surface and spherical cap
2
c = 2  h2r−h
V=


Spherical segment (spherical cap)
c
h
r
d
 

h
c2 h 2
= h

3
8 6
c2
2
A = 2 r h = 
h
4
c = 2  h 2r−h
c 24h2
r=
8h
V = h 2 r −



Spherical zone
c1
h
c2
d
r


 3 2 2
c c h 2
6 4 1 2
A = 2 r h = area of spherical surface
V=

2

2
2
2 2
c 2 c 2−c 1−4h
r=

4
8h

Lune
d
θ
The lune is bounded by two semicircles that are on a common diameter of the sphere; they are
separated by the dihedral angle . If r = d /2 is the diameter of the sphere, then


 4 3
2
1
r =  r 3 =
d 3
2 3
3
12
1 2
 
2
2
A= A=
4r  = 2 r = d
2
2
V=
Ellipsoid
b
a
c
V=
4
a b c
3
Note a, b, and c are the semidiameters, not the diameters.
The ellipsoid's equation is
x 2 y 2 z2
  =1
a 2 b 2 c2
Spheroid
[perry:2-7]
Prolate spheroid: formed by rotating an ellipse about its major axis 2a; minor axis is 2b:
[
A = surface area = 2 b 2
V=
ab −1
sin e
e
]
4
ab 2
3

2

b
e = eccentricity = 1−
a
Oblate spheroid: formed by rotating an ellipse about its minor axis 2b; major axis is 2a:
[
A = surface area =  2a 2
V=
b 2 1e
ln
e 1−e
]
4
a 2 b
3
Paraboloidal segment
The volume is shown in a side view:
h
r1
r2
k
d
The paraboloid can have 1 or 2 bases. If 1 base, the length is h, the radius is r 1 and d = 2r 1.
1
2
 r 1h
2
1
V for two bases = k r 21r 22
2
V for one base =
2
A = surface area for one base =
3p
p=
d2
8h
[  ]
d2
p2
4
3 /2
−p 3
Torus
R
r
D
d
2
Dd 2 = 2.4674Dd 2
4
2
2
A = 4  R r = 39.478 R r =  Dd = 9.8696 Dd
V = 2 2 R r 2 = 19.739R r 2 =
Barrel
h
d
D
The following are from [marks:2-22]. If the sides are bent to the arc of a circle, the approximate
volume is
V=
1
2
2
2
2
 h2D d  = 0.262h2D d 
12
If the sides are bent to the arc of a parabola, the exact volume is
V=
[
1
3
h 2D2Dd d 2
15
4
]
The number of gallons in a partially-filled cask standing on end is approximately 0.0034 n2 h where
1
n is number of inches in the mean diameter (i.e., Dd ) and h is the height of the liquid level in
2
−7 2
inches. If n and h are measured in mm and the volume in liters, the formula is 7.85×10 n h .
Solid or surface of revolution
Axis
r
G
Area = A
Theorem of Pappus: suppose a plane area A is rotated about an axis in its plane that doesn't
cross the area. The distance r is to the center of mass of the area. Then the volume of the figure
generated for a rotation of the area by an angle  is
V = As
where s = r  is the distance along the arc that the center of gravity traveled. For a full revolution,
the volume is thus
V = 2 r A
If the area is replaced by an arc of length L in the same plane and the arc is rotated by an angle 
around the axis, the resulting area of the generated surface is
A = Ls
where again s = r  is the distance along the arc that the center of gravity traveled. For a full
revolution, the surface area is thus
A = 2 r L
As an easy check, suppose we have a line of length h parallel to the axis of rotation and a distance
r from it. Then the surface area of the resulting cylinder is 2 r h. Note you can rotate the line
about its center of mass G about an axis that passes through G perpendicular to the plane and the
surface area remains constant.
The generalized Theorem of Pappus is: suppose you have any curved path of length s. If a
plane figure of area A or a plane curve of length L move so that its center of gravity slides along the
curved path while the area or the curve remain perpendicular to the path, then the volume
generated is As and the area generated is Ls. The path is assumed to curve slowly enough that
successive positions of A or L will not intersect.
Suppose a curve is rotated around the x-axis; the curve is defined by a regular function, parametric
equations, and a polar equation:
y = f x 
x = u t 
y = v t 
r = r 
When this curve is rotated one revolution around the x-axis, the surface area of the resulting
surface is [bartsch:371]

x2
x1
t2
2
 
   
  
dy
dx
A = 2 ∫ y 1
dx
2
= 2 ∫ v t 
t1
2
du
dv

dt
dt
2
dt
2
= 2 ∫ r sin 1
1
dr
d
d
When this curve is rotated one revolution around the x-axis, the volume of the resulting volume of
revolution is
x2
V = ∫ y 2 dx
x1
t2
= ∫ v 2
t1
2
dx
dt
dt
2
2
= ∫ r sin 
1


dr
cos−r sin d 
d
When y = f x  is rotated around the y axis, the volume is
y2
V = ∫ g y 2 dy
y1
x2
= ∫ x 2 ẏ dx
x1
where y = f x  and x = g y 
When r = r   is rotated around the y axis, the volume is
2
V = ∫ r 2 cos 2 
1


dr
sinr cos  d 
d
The method of shells sometimes makes it easier to calculate a volume gotten from rotating a
function y = f x  between x =a and x =b about the y axis
b
V = 2 ∫ x f  x dx
a
Cavalieri's theorem
[marks:2-23] Assume two solids have their bases in the same plane. If the plane section of one
solid at every distance x above the plane is equal in area to the plane section of the other solid at
height x above the base, then the two solids have equal volumes.
Plane geometry
Determining a right triangle
Let the length of the sides of the triangle be a, b, c, where c is the length of the longest side. Then
2
2
calculate  = a b . Classify the triangle by
 = c2
  c2
  c2
Right triangle
Acute triangle
Obtuse triangle
The classification using the first equality is the converse of the Pythagorean theorem.
Area of a triangle
If a, b, and c are the vectors of the sides of a triangle, then [wilson:81:108]
abc = 0
and the area A of the triangle is [wilson:81:108]
2 A = ∣a×b∣ = ∣b×c∣ = ∣a×c∣ =
2
2
a sinB sinC
b sin A sinC c sin A sinB
=
=
sin A
sin B
sinC
2
These relationships hold in three dimensions also as long as the vectors form a closed polygon.
Inscribed circle of a triangle
Given the vertex points P 1  x 1 , y 1, P 2 x 2 , y 2 , and P 3 x 3 , y 3 of a triangle, the location of the center
of the inscribed circle is
x , y  =

ax 1bx 2 cx 3 ay 1by 2cy 3
,
abc
abc

where a is the length of P 2 P 3, b is the length of P 1 P 3 , and c is the length of P 1 P 2. See the
General triangle section for the radius of the inscribed circle.
Circumscribed circle of a triangle
Given the vertex points P 1  x 1 , y 1, P 2 x 2 , y 2 , and P 3 x 3 , y 3 of a triangle, the location of the center
of the inscribed circle is
x , y  =
∣
∣∣
x 21y 21 y 1 1
x 22y 22 y 2 1
x 23y 23 y 3 1
D
,
∣
x 1 x 12y 21 1
x 2 x 22y 22 1
x 3 x 32y 23 1
D
∣ ∣
x1 y1 1
D = 2 x2 y2 1
x3 y3 1
See the General triangle section for the radius of the inscribed circle.
Equation of a line
Given (x1, y1) and (x2, y2) on a line, the equation of the line is [schmall:49:64]
y −y 1
y −y
= 2 1
x−x 1
x 2−x 1
which can be written
∣
x
x1
x2
y
y1
y2
∣
1
1 =0
1
The symmetrical form in terms of the intercepts of the axes is [schmall:50:65]
x y
 =1
a b
With a slope m and the y intercept of b, the equation is
y = mx b
The slope m is the tangent of the angle between the line and the x axis. The slope of the line
1
perpendicular to this line is − .
m
The point-slope form is
y −y 1 = m x −x 1 
The normal form of the equation is [schmall:53:68]
x cosy sin = p
where p is the distance of the line from the origin and  is the angle of the perpendicular to the line
from the x axis. The normal's angle with the x axis will be from /2 to 3/2.
y

p
x
This can be rewritten as
y =−
which shows that the y intercept is b =
x
p

tan  sin
p
.
sin 
Any first degree equation in two variables represents a straight line
Ax ByC = 0
These are obvious simplifications of the corresponding equations for three dimensions.
Polar coordinates
By converting the slope-intercept form of the equation of a line, we get the polar form
sin−m cos = b
In polar coordinates, the equation of a straight line between two points (1, 1) and (2, 2) is
[schmall:66:81]
sin1−2  sin2 − sin−1 


=0

1
2
or
∣ ∣
1

1
1
1
2
cos
sin 
cos 1 sin1 = 0
cos 2 sin2
Angle between two lines
If y = m1x + b1 and y = m2x + b2, we have [schmall:60:75]
tan  =
m1−m2
1m 1 m 2
Distance from a line to a point
If the equation of the line is x cosy sin  = p , the distance from the point r1 is [schmall:63:78]
∣x 1 cos y 1 sin−p∣
If the equation of the line is Ax + By + C = 0 and the point is (x, y), the distance is [schmall:63:78]
d=
AxBy C
 A2B2
Bisectors of angles between two lines
Given the lines Ax + By + C = 0 and A1x + B1y + C1 = 0. The point (x, y) on the bisector is
[schmall:64:79]
Ax By C
 A B
2
2
=±
A1 x B1 y C 1
 A B
2
1
2
1
The + sign is used if the bisector and the origin lies within the same one of the four possible angles.
If the equations of the lines are
x cosy sin = p
x cos1 y sin1 = p1
then the equation of the bisector is
x cos1±cos y sin 1±sin = p 1±p
Equation of circle
A circle with center at (x0, y0) and radius r has the equation
x −x 0 2y −y 0 2 = r 2
Every equation of the form x2 + y2 + 2ax + 2by + c = 0 represents a circle with center at (-a, -b) and
2
2
radius  a b −c [schmall:87:102].
The equation of the tangent line to a circle at point x 1, y 1  is [schmall:321:355]
x x 1y y 1 = r 2
Polar equation of circle
If the circle's center is (1, 1) and the radius is r, then the equation is [schmall:117:131]
 2−2  1 cos−1  21−r 2 = 0
For the circle x2 + y2 + 2Gx + 2Fy + C = 0, the polar form is [schmall:119:134]
 22G cos F sin = 0
where the pole is on the circle (i.e., C = 0).
Equation of parabola
y
P(x, y)
M
Directrix
R
p
O
x
F(a, 0)
L
2
The equation is y =4ax . Because FP = MP is always true and =1 for the parabola, the focus is
at a distance a from the origin on the x axis, the distance p = a. For any two points x 1 , y 1 and
y2
x
x 2 , y 2  on the parabola, we have 12 = 1 . The tangent at P bisects the angle FPM. LR is the
x2
y2
latus rectum and its length is 4a.
The equation of the tangent line to the parabola at x 1 , y 1 is [schmall:137:152]
y y 1 = 2a xx 1
Given a point x 1 , y 1 we have that there is two distinct, one, or no lines tangent to the parabola
corresponding to whether the point is outside, on, or inside of the parabola. The slope(s) of the
line(s) are [schmall:143:158]
2
y 1± y 1−4ax1
m=
2x1
Two tangents to the parabola that are perpendicular to each other intersect on the directrix
[schmall:144:159].
The line y = mx 
a
is tangent to the parabola for all nonzero values of m [schmall:138:153].
m
The polar equation for a right-opening parabola with the origin at the focus is [schmall:153:168]
2a
1−cos 
r=
With the origin at the vertex, the polar equation is [schmall:153:168]
4a cos 
sin2 
r=
Equation of ellipse
C
R
b
a
a
B
A
O
F1
F2
P
Figure 2
Also see the section Ellipse on page 107. [http://en.wikipedia.org/wiki/Ellipse]
F 1 and F 2 are called the foci and have the property that if a light ray is emitted at one foci, it will be
reflected to the other foci if the inside of the ellipse is perfectly reflecting. It is also true that the
distance F 1 PF 2 P is a constant for any point P on the ellipse.
The general equation of an ellipse is
Ax 2 Bxy Cy 2DxEy F = 0
2
provided F ≠0 and F B −4AC   0 ; this ellipse is at an angle to the axes. If B = 0 and AC > 0,
then the ellipse's axes are parallel to the coordinate axes.
Another form is
Ax 2 Bxy Cy 2DxEy = 1
with B 2−4AC  0.
By a translation and rotation, the ellipse's equation can be put in canonical form (ellipse's center at
origin and the major diameter lies along the x axis):
2
2
 
x
y

a
b
=1
The major diameter is 2a and the minor diameter is 2b. This equation can also be written

2

x
y =±b 1−
a
=± a −x 1− 
2
2
2
The foci are (-a, 0) and (a, 0) where  is the eccentricity

2

 = 1−
b
a
A point is outside, on, or inside the ellipse if x /a 2 y /b 2 is > 1, = 1, or < 1, respectively.
The equation of a tangent line with nonzero slope m to the ellipse is y = mx ±  am  b 2 .
2
2b 2
.
a
2
2
The distance from the center to any focus is  a −b . The dashed line is the directrix (there's one
a
at the other end of the ellipse also) and is a distance from the center.

The distance to the directrixes from the origin is a/. F 1 R is called the latus rectum and is
An ellipse centered at (h, k) and with the major axis parallel to the x axis has the equation
 x−h2 y −k  2

=1
2
2
a
b
which can also be given in parametric form (− ≤ t ≤ )
x = ha cos t
y = k b sint
A general parametric form is
x = x 0a cost cos −b sint sin
y = y 0a cos t sin−b sint cos
where 0 ≤ t ≤ 2
(x0, y0) is the center of the ellipse and  is the angle between the x axis and the major diameter axis
of the ellipse. For an ellipse in canonical position (center at origin, major diameter along the x
axis), the equation is
x = a cos t
y = b sint
In polar coordinates with the origin at the center of the ellipse and the polar angle  measured from
the major diameter axis, the equation is
r=
ab
b
=
2
2
a sin b cos   1−2 cos2 
If instead the origin is translated to a focus, then the other focus is rotated to an angle , the polar
equation is
r=
a1−
1±cos −
The general polar form is
r=
PQ
R
where
P = r 0 [ b 2−a 2cos0 −2a 2b 2cos−0 ]
Q =  2 ab  R−2r 20 sin2 −0 
2
2
2
2
R = b −a cos2 −2 a b
and the center of the ellipse is at (r0, 0), the major diameter is 2a, the minor diameter is 2b, and the
major diameter axis is rotated by  relative to the polar axis.
The circumference of the ellipse is 4aE where E is the complete elliptic integral of the second
kind. A power series is
{[
2
]
[    
∞
C = 2 a ∑ −
n=0
= 2 a 1−
1
2
n
∏
m=1

2m−1
2m
2
2−
1⋅3
2⋅4
2n

2n−1
}
2 4

1⋅3⋅5
−
3
2⋅4⋅6
2

6
−
5
]
Equation of hyperbola
directrix
M
P
b
a
D
F
O
The basic property of the hyperbola is that MP =  FP . The equation of the asymptotes are
b
y =± x .
a
The hyperbola is represented by
Ax 2 Bxy Cy 2 DxEy F = 0
2
where B  4AC . For the hyperbolas opening in the x direction shown in the figure, the equation is
x 2−y 2 = 1 . For hyperbolas opening in the y direction, the equation is y 2−x 2 = 1 . From here on,
the formulas are for a hyperbola opening in the x direction. For a hyperbola centered at (h, k), the
equation is
 x−h2 y −k  2
−
=1
2
2
a
b

2
The eccentricity is  = 1
located at x = a/.
2
The polar equation is r =
equation is r =

b
2
2
2
. The foci are at (h + c, k) where c = a b . The directrix is
a
a
where a is a constant. If the origin is at the left focus, the
cos 2
a 2 −1
[schmall:216:231].
cos −1
Parametric equations are
a
h
cos t
y = b tan t k
x=
x =±a cosh th
y =b sinh tk
or
The equation of the tangent to the hyperbola at x 1 , y x  is [schmall:202:217]
x x1
2
a
−
y y1
b2
=1
The line y = m x ±ma −b with nonzero slope m is tangent to the hyperbola for all values of m
2
2
[schmall:203:218]. The tangent lines to the conjugate hyperbola are y = m x ± b −ma .
2
2
Given the asymptotes A1 x B1 y C 1 = 0 and A2 x B 2 y C 2 = 0, the equation of the hyperbola is
 A1 x B 1 y C1  A2 xB 2 y C 2  = k where k is some constant [schmall:211:226]. The area of the
triangle formed by any tangent and the asymptotes is constant [schmall:216:231].
Conic sections
http://en.wikipedia.org/wiki/Conic_section
A conic section is the intersection of a plane with a right circular cone. The conic section is
represented by the general algebraic equation of degree 2. Conic sections have the geometrical
property that the horizontal distance DP is equal to the distance FP for any point P on the conic
section. The point F is the focus.
Conics have polar equation r = L /1± cos where L is the semilatus rectum and  is the
eccentricity.
M
P
D
F
E
O
G
R
Directrix
Equation‡
Name
2
2

2
circle
x y =a
0
ellipse
x /a 2 y /b 2=1
y 2 = 4ax
parabola
‡
0
 1−b/ a2  a 2−b 2
1
hyperbola x /a 2− y /b 2=1
c = FO L = FR
a
 1b/ a2  a 2b 2
p = EF
a
∞
b 2 /a
b /  a −b
2a
2a
b 2 /a
b /  a b
2
2
2
2
2
2
The origin is at the vertex.
Note the relations p  = L and ae = c .
For the case where the origin is at a focus, we have ( = a 2b 2,  = a 2−b 2)
Name
x‡
y‡
Polar
Cartesian
r =a
x 2y 2=a2
circle
a cos t a sin t
ellipse
a cos t b sin t b 2 /a−  cos   x − /a2y / b 2 = 1
parabola
at2
2at
2a /1−cos
y 2 = 4a x a
hyperbola a /cost b tant b 2 /a−  cos x  / a2−y /b 2 = 1
‡
Parametric equations.
Five points determine a conic if no three are collinear; i.e., there is one unique conic passing
through them.
0 < e < 1 ellipse
e =1 parabola
e =0 circle
e >1 hyperbola
Directrix
2
2
The equation Ax Bxy Cy DxEy F = 0 can be rearranged to give
Ax 2 Bxy Cy 2 =−Dx Ey F 
2
2
which shows that the quadratic form z = Ax Bxy Cy and the plane z =−Dx Ey F 
intersect to yield a conic section. Parabolas and hyperbolas are gotten by a horizontal plane
D = E =0 while ellipses require the plane to intersect the x-y plane at an angle. Degenerate
conics correspond to degenerate intersections.
Conic sections with matrix notation
http://en.wikipedia.org/wiki/Matrix_representation_of_conic_sections
Conic sections in the Cartesian plane are represented by the equation
ax 2bxy cy 2dxey f = 0
or in homogeneous coordinates as rTQr = 0 where Q is the symmetric matrix

x
r= y
1
Let M =





a b /2 d / 2
and Q= b /2
c e /2
d / 2 e /2 f

c e/2
a b/2
, and E = det(N).
, D = det(M), N =
e/
2 f
b/ 2 c
If D = 0, the conic is degenerate:
D  0 : two intersecting lines
D = 0 : two (possibly coincident) parallel straight lines
D  0 : empty
If D is not zero, the conics can be classified by the determinant of the minor Q11:
E  0 : hyperbola
E = 0 : parabola
E  0 : ellipse (a circle if Q 11=Q 22
The center of the conic is
be−2cd
4D
db−2ae
yc =
4D
xc =
The major and minor axes pass through the center of the conic and their directions are the
eigenvectors of M.
Through a point p, there are generally two tangents. The points of tangency are the intersection of
the conic with the line pTQr = 0. When p is on the conic, the line is tangent there. When p is inside
an ellipse, the line is the set of all points whose own associated line passes through p.
To write the canonical form of the equation of the conic, the conic is translated and rotated so that
the conic's center is at the origin and its axes are parallel to the coordinate axes. Let 1 and 2 be
the eigenvalues of M. We have
2
2
1 x ' 2 y ' 
D
=0
det M 
where the coordinates are
y'
x'
y
x
Divide through by D/det(M) to get the canonical form.
Circle through three points
For each point, write the equation x2 + y2 + 2Gx + 2Fy + C = 0. This represents three equations in
three unknowns G, F, and C. Solve for them. Equivalently, use the equation (x - h)2 + (y - k)2 = r2
and solve for the constants.
For the three points (x1, y1), (x2, y2), (x3, y3), the equation is [schmall:97:112]
∣
2
2
∣
x y
x y 1
2
2
x 1 y 1 x 1 y 1 1
=0
x 22 y 22 x 2 y 2 1
x 23 y 23 x 3 y 3 1
Tangent to a circle
Given a circle x2 + y2 = r2, the equation of the tangent at (x1, y1) is [schmall:89:104]
xx 1yy 1 = r 2
For the circle x2 + y2 + 2Gx + 2Fy + C = 0, the equation of the tangent at (x1, y1) is [schmall:94:109]
xx 1yy 1G x x 1 F y y 1C = 0
For the circle (x - x0)2 + (y - y0)2 = r2, the equation of the tangent at (x1, y1) is [schmall:96:111]
x 1−x 0 x −x 0 y 1−y 0 y −y 0  = r 2
For the line y = mx + b intersecting the circle x2 + y2 = r2, the line
y =mx ±r  1m
2
is tangent to the circle for all values of m [schmall:98:113].
For the line y = mx + b intersecting the circle x2 + y2 + 2Gx + 2Fy + C = 0, the equation of the
tangent is [schmall:96:113]
2
2
y F = m x G ± G F −C⋅ 1m
2
The ± is used because there will be two tangents:
If we have a point P = (x1, y1) and a circle (x - x0)2 + (y - y0)2 = r2, the length L of the tangent from the
point P to the point of tangency is [schmall:98]
L 2 = x 1−x 0 2  y 1−y 0 2−r 2
(x0,y0)
L
(x1,y1)
If the equation was x2 + y2 + 2Gx + 2Fy + C = 0, the tangent's length L is [schmall:99:114]
L 2 = x 21y 212Gx 12Fy1C
Angle of intersection of two circles

P
A
B
Let [schmall:110:125]
AB = 
AP = r 1
BP = r 2
Suppose the equations of the two circles are
x 2 y 22G 1 x 2F1 y C 1 = 0
2
2
x y 2G 2 x 2F2 y C 2 = 0
Then
cos =
2G1 G 22 F 1 F 2−C 1−C2
2r1r2
The condition that the two circles cut orthogonally is
2G1 G 22F 1 F 2−C 1−C 2 = 0
Center of triangle's inscribed circle
For a triangle with vertices (x1, y1), (x2, y2), (x3, y3), the center of the inscribed circle is
[schmall:122:137]

ax 1bx 2cx 3 ay 1by 2cy 3
,
abc
abc

where a, b, c are the lengths of the triangle's sides.
Euclidean Transformations
The relevant group is E(2), comprised of rotations and translations. As usual, translations and
rotations do not commute, as shown by inspection of the infinitesimal generators ∂ x, ∂ y, and
x ∂y −y ∂x .
Translations
To transform from (x, y) to (X, Y) with a translation to (x0, y0), the transformation equations are
X = x x 0
Y = y y 0
The inverse transformation is
x = X −x 0
y = Y −y 0
In polar coordinates, suppose we have the new (translated) origin at r 0 , 0 , where r and  are in
the original coordinate system. The new coordinate system's coordinates will be denoted by
carets. The old coordinates are related to the coordinates in the new system by [lawrence:17]:
r 2 = r 02r 22 r r cos0−
 0 sin0
r sin r
tan =
 0 cos0
r cos r
Rotations
If the axes are rotated by an angle , the transformation equations can be used to find the old and
new coordinates of a point P (which doesn't rotate). If the old coordinates are (x, y), use the
following transformation to get things in terms of the new coordinates
x = X cos−Y sin
y = X sinY cos 
The inverse transformation is
X = x cos y sin
Y = −x siny cos 
Note the transformation is orthogonal: the inverse transformation matrix is the transpose of the
forward transformation matrix.
Oblique transformations
Suppose the positive direction of the new x axis (call it X) makes an angle  with the old x axis;
analogously, the new y axis (call it Y) makes an angle  with the old x axis. Then we have the
transformation [schmall:126:141]
x = X cosY cos 
y = X sinY sin 
See the section Three noncoplanar vectors as a base (reciprocal system) on page 73 for the
extension to three dimensions.
Oblique coordinates
If the x and y axes are separated by an angle  and P1 = (x1, y1) and P2 = (x2, y2) are any two
points, then [crc:252]
Distance between P1 and P2:
  x −x
1
2
2
2
  y 1−y 2 2 x 1−x 2  y 1− y 2cos 
Point dividing P1P2 in ratio r/s:

rx 2sx 1 ry 2sy 1
,
r s
r s
Midpoint of P1P2:

Area of triangle P1P2P3:
x 1x 2 y 1 y 2
2,
2


1
sin x 1 y 2x 2 y 3 x 3 y 1−y 1 x 2 −y 2 x 3−y 3 x 1 
2
Circle: center at (h, k), radius r:
x −h2y −k 22 x−hy −k cos = r 2
Archimedean spiral


2
 
dr
a
2
Polar equation r = a . Arc length s  = ∫ r 2
d  = [ ln  ] where  =  1 .
d

2
0
Note ln  = sinh−1 .
Curvature
For a function y = f x , the curvature is [lawrence:22]
K=
u xx x 
 1[u x x ]2 
3/ 2
The radius of curvature is  = 1/∣K∣.
If the curve is given in parametric form x = f t  and y = g  t , the curvature is
K=
f˙ g̈−f̈ ġ
 ḟ 2ġ 2 
3/2
If the curve is f x , y  = 0, the curvature is
K=
f xx f 2y −2f xy f x f y f yy f 2x
 f 2x f 2y 
3 /2
In polar coordinates, the curvature is
K=
where r˙ =
r 22 ṙ 2−r r̈
3 /2
 r 2 ṙ 2 
dr
.
d
The center of curvature is the point  ,  that is the center of the circle tangent to the curve at
x , y  with the indicated radius of curvature. In Cartesian coordinates:
ẏ  1 ẏ 2 
ÿ
1 ẏ 2
 = y
ÿ
 = x−
In parametric form:
 = f − ġ
 = g ḟ
2
2
ḟ ġ
=
ḟ g̈−f̈ ġ
In polar coordinates:
 = r cos −  r 2 ṙ 2 d r sin
 = r sin   r 2 ṙ 2 d r cos
 =  r 22 r˙ 2−r r¨ 
Products
Pseudovectors and pseudoscalars
Pseudovectors and pseudoscalars change signs under parity transformations (a reflection where
one coordinate changes sign).
The cross product is a pseudovector. The dot product of two vectors is a scalar; the dot product of
a vector and a pseudovector is a pseudoscalar. Thus, the scalar triple product is a pseudoscalar.
Scalar (dot) product
a⋅b ≝ ab cos 
where  is the smallest angle between the vectors. You can see 0 ≤ ∣a⋅b∣ ≤ ab and a⋅b ∈ ℝ . This
form emphasizes the invariant nature of the scalar product under Euclidean group coordinate
transformations‡ because it depends only on the magnitudes of the vectors and the angle between
them. Properties:
a⋅b = b⋅a
a⋅bc  = bc⋅a = a⋅ba⋅c
 a⋅b = a⋅b
2
a⋅a = a
Two nonzero vectors a and b are perpendicular iff a⋅b = 0 . If a⋅a = 0 , then a must be the zero
vector.
In terms of components:
a⋅b = a x b x a y b y az b z
Suppose a, b, and c are vectors that make a closed triangle. Then we can get the cosine law:
abc = 0 and
a−b⋅a−b  ≝ a−b2 = a 2b 2−2ab cos
The geometric picture of the dot product is that it yields the projection of one vector along another:
a
a▪b
θ
b
If a represents a plane area and if b is a vector inclined to that plane, then a⋅b is the volume of the
slanted cylinder as shown in the following figure [wilson:57:84]
‡ See the comments on invariance in the Invariance section.
h
b
θ
a
This is because a is parallel to the vertical axis and the altitude h is b cos .
Vector (cross) product
c = a×b
c =ab sin
c i = ijk a j b k
where the direction of c is found by the usual right hand rule and  is the angle between a and b.
Again, this is an invariant with respect to Euclidean group coordinate transformations‡, but c is a
pseudovector because of its behavior with respect to parity transformations. If three vectors form a
triangle (closed polygon), then the cross product between any two of the sides is twice the area of
the triangle. The summation convention in the last form is used over the three space indices.
c
b
A = area
a
We have 2A = ∣a×b∣ = ∣b×c∣ = ∣a×c∣. Note ∣a×b∣ is the area of the parallelogram formed by a and
b.
Properties:
a×b = −b×a
ab ×c = a×c b×c
a×a = 0
i × j = k
j ×k = i
k ×i = j
d a×b = d a×ba×d b 
a×b×cb×c×ac×a×b = 0
a×bc×d = a−c ×b−d a×d c×b
a×b⋅c ×d  = a⋅c b⋅d −a⋅d b⋅c
d
a×b = ȧ×ba×ḃ
dt
Note a×b = a×c does not imply b = c, but rather a×b−c  = 0.
If M is a 3x3 matrix, the cross product obeys the following identity:
‡ See the comments on invariance under the Invariance section.
T
 M a × M b  = det M   M −1  a×b
Two nonzero vectors a and b are parallel iff their cross product vanishes.
If a and b are the sides of a parallelogram, a×b is the area of the parallelogram.
In terms of components:
a×bx = ay bz −a z b y
a×by = a z b x −ax b z
a×b z = a x b y −a y b x
which can be remembered by symbolically expanding the determinant along the first row:
∣
∣
i
j k
a×b = a x a y az
bx by bz
The component of a perpendicular to b is
b⊥ =
a×b×a
a⋅a
and the component parallel is b∥ = a⋅b .
The Pythagorean theorem relates the cross and dot products:
2
∣a×b∣2 = a 2 b 2− a⋅b  = a 2 b 2  1−cos2  =  a b sin 
2
where  is the angle between a and b.
Scalar triple product
∣
∣
ax ay
[a b c ] ≝ a⋅b×c = b x b y
cx cy
i
az
bz
cz
j
= abc cos sin = ijk a b c
k
where  is the angle between b and c and  is the angle between a and b×c. This quantity is
invariant under coordinate rotation (it's a pseudoscalar because the sign can change under a
coordinate system parity change). The absolute value is the volume of the parallelopiped whose
edges are a, b, c.
The three nonzero vectors are coplanar iff a⋅b×c = 0.
Other properties:
a⋅b×c a = a×b ×a×c
a⋅b×c = b⋅c×a = c⋅a×b
a⋅b×c = −a⋅c ×b
d
a⋅b×c  = ȧ⋅b×ca⋅ḃ×c a⋅b×ċ
dt
In the scalar triple product, the dot and cross can be interchanged and the components permuted
cyclically; a change of cyclic order changes the sign. This is most easily seen by virtue of the
interpretation as the volume of a parallelopiped.
Four vectors have a linear relationship [wilson:76:103]:
[a b c ] d [c d a] b = [b c d ] a[d a b ] c
Another way of saying and writing this is that any vector r may be represented in terms of three
others a, b, c by [coffin:232:261]
r [a b c ] = [r b c ] a[ r c a ] b[r a b ] c
A reduction formula [wilson:86:113]
∣
a⋅d
[a b c ][d e f ] = b⋅d
c⋅d
a⋅e a⋅f
b⋅e b⋅f
c⋅e c⋅f
∣
Other reduction formulas [wilson:113:139]
a×b×c ×d  = [a c d ] b−a⋅b c×d = b⋅d a×c−b⋅c a×d
[ a×b b×c c×a] = [ a b c ]⋅[ a b c ]
Three noncoplanar vectors as a base (reciprocal system)
The scalar triple product is of use in expressing a given vector in terms of three noncoplanar
vectors a, b, c [wilson:81:108]. Let
r = a ab bc c
By multiplying by ⋅b×c and the other corresponding terms, we find
r=
[r b c]
[r c a]
[ r a b]
a
b
c
[a b c ]
[b c a ]
[c a b ]
We can define
A=
b×c
c ×a
a×b
, B=
, C=
[a b c]
[a b c ]
[a b c ]
These vectors constitute the reciprocal base. Note a⋅A = b⋅B = c⋅C = 1 and all other dot products
between the two sets are zero (this is easy to recognize because of the cross products in the
definitions). (A, B, C) is reciprocal to (a, b, c) iff these dot product relations hold[wilson:85:112].
Then the expression for the arbitrary vector r is
r = r⋅Aar⋅B br⋅C c
These represent a concise formulation to convert between Cartesian and oblique coordinate
systems in three dimensional space. We can also express r in the reciprocal base as
r = r⋅a Ar⋅b Br⋅c C
The previous two equations can be used to write down the transformation and its inverse between
the two coordinate systems. Note  i , j , k  is its own reciprocal system; this is the only basis for
which this is true (along with the equivalent left-handed basis) [wilson:87:114].
These reciprocal vectors are used in crystallography to discuss reciprocal lattices (see
http://en.wikipedia.org/wiki/Reciprocal_lattice). A reciprocal lattice vector k has the property that
e 2 i k⋅r = 1 for all lattice point positions r; i.e., k⋅r is an integer.
If (A, B, C) is reciprocal to (a, b, c), then
[ A B C ][ a b c ] = 1
Vector triple product
The definition is d = a×b×c . The parentheses are necessary because the cross product is
nonassociative: a×b×c  ≠ a×b×c . We have [wilson:76:103] [wilson:111:137]
[wilson:113:139]
a×b×c  = b a⋅c −c a⋅b
a×b ×c = a⋅c b−b⋅c a
a×b×c = −c×a×b 
a×a×b  = a⋅b a−a2 b
a×b×c b×c×ac×a×b = 0
a×b×c ×d  = [ a c d ] b−a⋅b c ×d
= b⋅d a×c−b⋅c a×d
The first equation yields the "bac cab" mnemonic to help remember the formula.
The vector triple product is a vector perpendicular to a lying in the plane of b and c.
In terms of Levi-Civita
a×b×c i = ijk a j klm b l c m = ijk klm a j b l c m
The vector d is perpendicular to a and b×c; since b×c is perpendicular to the plane of b and c, d
must lie in the plane of b and c and thus takes the form
a×b×c  = s bt c
s , t ∈ℝ
and, from the rule above, s = a⋅c and t = −a⋅b .
Also [wilson:76:103]
∣
∣
a⋅c a⋅d
b⋅c b⋅d
a×b×c×d  = [b c d ] a−[c d a] b[ d a b] c−[a b c ] d
a×b⋅c×d  = a⋅c b⋅d −a⋅d b⋅c =
An arbitrary combination of dot and cross products can be broken into a sum of terms each
involving only one cross product [salmon:77:104].
Direction cosines
A direction in space can be specified using direction cosines. Suppose a vector r points in the
desired direction. Then the direction cosines are
r
r
r⋅i
r⋅j r y
r⋅k
=
= x
=
=
=
= z
r
r
r
r
r
r
We will use  =  ,  ,  to indicate the vector made up of three direction cosines. Note ∣∣ = 1 .
Direction numbers are   where  is nonzero.
Suppose  1 = 1 , 1 , 1 and  2 = 2 , 2 , 2 . These two directions are parallel iff  1 =  2 .
These directions are perpendicular iff  1⋅ 2 = 0 . [drmath]
Three directions are parallel to a common plane iff [drmath]
∣
∣
1 1 1
 2  2 2 = 0
3 3 3
If we are given two lines in space with direction cosines  1 and  2 we can calculate the direction
cosines of the line that is perpendicular to both lines from [snyder:24:44]



±1
=
=
=
1 2−2 1 1 2−2 1 1 2−2 1 sin
where  is the angle between the two lines [snyder:8:28]
cos =  1⋅ 2
and
sin2  = 1− 1⋅ 2 =  1 × 2 2
where the square of a vector a is a⋅a (this relation can be proved by writing
1− 1⋅ 2 =  21  22− 1⋅ 2 and expanding the right hand side in components).
Equation of a line
Forms
Vector
If r0 is a point on the line and a vector v is parallel to the line, then all points r on the line can be
found from [corral]
r = r 0t v
t ∈ℝ
Endpoints of two vectors
Given two vectors r1 and r2, the equation for the line between the ends of the two vectors is
r = t r 11−t r 2
Parametric
This form is composed of the components of the vector form:
x = x 0at ,
y = y 0bt ,
z = z 0ct
where v = (a, b, c).
Symmetric
By solving the parametric equations for the parameter and equating in pairs, we get [snyder:20:41]
x −x 0
y −y 0 z−z 0
=
=
a
b
c
If, e.g., a = 0, we can't divide by zero, but the parametric form gives x = x0. This means the line lies
in the plane x = x0. a, b, and c are direction numbers of the line and can be converted to direction
cosines by dividing by  a 2b 2c 2 .
Two-point form
Using a second point r1 with the symmetric form, we have [snyder:20:41][drmath]
x −x 0
y −y 0
z−z 0
=
=
x 1−x 0
y 1−y 0 z 1 −z 0
Equation from two points on line
If r1 and r2 are points on the line, then all points on the line can be found from [corral]
r 1t r 2−r 1 
t ∈ℝ
(the previous equation is the same as that given in section Endpoints of two vectors above. The
parametric form is
x = x 1x 2−x 1t ,
and the symmetric representation is
y = y 1 y 2−y 1  t ,
z = z 1 z 2−z 1 t
x−x 1
y −y 1
z −z 1
=
=
x 2−x 1
y 2−y 1 z 2−z 1
From two planes
Two planes that intersect result in a line. This implies the line is the locus of points that satisfy the
two simultaneous equations [snyder]
A1 x B1 y C1 zD 1 = 0
A2 x B2 y C2 zD 2 = 0
See more details in the section Intersection of two planes on page 81.
Equation of a plane
General form
Every equation of the first degree in x, y, and z represents a plane
Ax By CzD = 0
or
r⋅Q = −D
if Q is (A, B, C). [wilson:88:115] Any scalar equation of the first degree in an unknown vector r may
be reduced to this form and, hence, any scalar equation of the first degree in r represents a plane.
Let
2
2
 =  A B C
2
If we divide the general equation by , we get
 x  y  z p = 0
where , , and  are the direction cosines of the normal to the plane and
p=
D

Hessian normal form
The preceding allows us to put the equation of the plane in the Hessian normal form:
 =p
n⋅r
where p is the perpendicular distance from the origin to the plane and n =  ,  ,  is a unit normal
to the plane. p can be positive or negative, depending on where the plane is with respect to the
origin. Thus, in words: the position vector of any point on the plane dotted with a unit normal
to the plane is a constant.
Here's a picture that illustrates the notion that any point in the plane dotted with a normal to the
plane is a constant [corral:35:43]
n = normal vector
to plane
R = r - r0
r = (x, y, z)
r = (x0, y0, z0)
Let r and r0 be any two points in the plane. Then R = r - r0. Since R lies in the plane, we have n·R
= n·(r - r0) = 0. Thus, n·r = n·r0. Since r and r0 are arbitrary points in the plane, we have that n·r is
the same constant for any point in the plane.
 = p . Note we're now using the unit normal n . Since we now know this is a constant, let's
Let n⋅r
look at the case where r is parallel to n -- this is the case when r is the perpendicular to the plane
from the origin. Thus we have r = p, so we see p is the distance from the origin to the plane. Since
it's a dot product, we could also have r and n be antiparallel, so p could be negative. You could
take the convention that the unit normal should always point towards the origin, so then p would
always be negative.
Here's a geometrical picture that illustrates this further. Draw the line OB perpendicular to the
plane AE from the origin O. The cross section will be:
B
D
A
C
|p|
E
r
O
The line OD represents any vector r in the plane AE. This vector dotted with the unit normal in the
direction OB (i.e., its projection onto the normal direction) is the vector OC. Since the point D is
arbitrary, you can see that n·r is constant for all points in the plane because you can rotate the line
AE about OB to get to all possible points in the plane.
Note that OC = ∣p∣.
Dual form
The plane can be represented by a single vector [wilson:108:135] p. The direction of p will be from
the origin perpendicularly to the plane and the magnitude will be the reciprocal of the length of that
perpendicular The equation of the plane in Hessian form is then
r⋅p = 1
The components of p are the reciprocals of the intercepts of the plane on the axes.
From axial intercepts
If a, b, and c are the intercepts on the x, y, and z axes, then the equation of the plane is
x y z
  =1
a b c
Through origin and parallel to two vectors
r = s r 1t r 2
where s and t are parameters.
From three noncollinear points
Let the points be r1, r2, and r3. Define
n = r 2−r 1 ×r 2−r 3 
If the three points are collinear, this vector is identically zero. Otherwise, it's a normal to the plane
containing the three points.
[coffin:232:261]:
r −r 1⋅r 1−r 2 ⋅r 2−r 3  = 0
or
⋅r −r 1  = 0
where
 = r 1×r 2r 2×r 3r 3 ×r 1
The condition that the points lie in the plane Ax By CzD = 0 is
∣
x y z
x 1 y 1 z1
x 2 y 2 z2
x 3 y 3 z3
∣
1
1
=0
1
1
Two parametric equation forms are
r = r 1s r 2−r 1t r 2−r 3
r = s r 1t r 2 1−s−t r 3
Alternative equations are [crc:356]
[r r 1
r = s r 1t r 2u r 3
or
[ r −r 1 r 1−r 2 r 2−r 3 ] = 0
or
r 2 ]  [r r 2 r 3 ]  [ r r 3 r 1 ] − [ r 1 r 2 r 3 ] = 0
where s ,t ,u ∈ℝ.
Through a point and perpendicular to a vector
The equation of a plane containing the point r0 = (x0, y0, z0) and perpendicular to the nonzero vector
n (i.e., n is a normal to the plane) is [corral:35:43]
n⋅r −r 0  = 0
Here, r = (x, y, z) is any point in the plane. This is seen in the following picture:
n
R
(x, y, z)
(x0, y0, z0)
For any point in the plane, n·R = 0 and R = r - r0. This can also be written as [crc:356]
r⋅n=c
where c is a constant and is n·r0. The perpendicular distance from any point r to this plane is
c −n⋅r
r
From two simultaneous equations: projecting planes
Suppose we have the simultaneous equations
x = m zb
y = n z c
are two planes which intersect in a straight line. They are perpendicular to the xz and yz planes,
respectively. These two planes are called projecting planes.
To find the projecting plane of a given line
Let the line be given by the Hessian forms
S 1 = n1⋅r −d 1 = 0
S 2 = n2⋅r −d 2 = 0
Then the equation
S 1S 2=0
represents any plane passing through the intersection of the above two planes. Choose  and  so
that one of the variables will vanish, then we'll have an equation for one of the projecting planes of
the given line. In practice, the desired variable is simply eliminated from the two equations.
Points
Collinear points
r1, r2, r3 are collinear iff [drmath]
x 2−x 1
y −y 1 z 2−z 1
= 2
=
x 3−x 1
y 3−y 1 z 3−z 2
Coplanar points
r1, r2, r3, r4 are coplanar iff [drmath]
∣
x1
x2
x3
x4
y1
y2
y3
y4
z1
z2
z3
z4
∣
1
1
=0
1
1
Lines and Planes
Point dividing a line segment into a given ratio
Suppose we have two points P1 = (x1, y1, z1) and P2 = (x2, y2, z2). We want to find the point P on the
line between P1 and P2 such that
PP 1 m 1
=
PP 2 m 2
The solution is [snyder:8:28]
m 2 x 1m 1 x 2
m 1m 2
m y m 1 y 2
y= 2 1
m 1m 2
m 2 z 1m 1 z 2
z=
m1 m 2
x=
Note if m1 and m2 have opposite signs, the point P lies outside the segment P1P2.
Lines in space
Suppose line L1 is given by r1 + sv1 and L2 is given by r2 + tv2. Then these lines are either
1. Identical
2. Parallel: v1 is parallel to v2.
3. Intersect
4. Skew: they don't intersect, but lie in parallel planes.
For the lines to be perpendicular, we must have v1 perpendicular to v2. Note that skew lines can be
perpendicular.
Line in a plane
Given a line
x = mza
y = nzb
and a plane Ax By CzD = 0. For the line to be in the plane, we must have [salmon:29:55]
AmBnC = 0
AaBbD = 0
(line is parallel to plane)
and
(one of the line's points is in the plane)
Line through a point parallel to a vector
We desire the equation of a line through the point r0 and parallel to a vector A. If r is the vector to
any point on the line, then r - r0 is parallel to A; thus, the vector product vanishes and we have
[wilson:107:134] [coffin:232:261]
A×r −r 0  = 0
Intersection of two lines
To determine intersection, use the parametric representation of the lines, then set the two x , y ,z 
triples equal [corral:34:42]. This will result in a system of three equations in two unknowns (the two
parameters).
For two non-parallel lines (i.e., r2 and r4 are not parallel)
r = r 1s r 2
r = r 3t r 4
a necessary and sufficient condition that the lines intersect is [crc:355]
[r 1−r 3 r 2 r 4 ] = r 1−r 3 ⋅r 2×r 4 = 0
The equation of the plane holding the two intersection lines is found from the normal, which is r2×r4
and the intersecting point.
If the two lines are given by the equations
x = mzb
x = m 1 zb1
y = nzc
y = n1 z c 1
then the lines will intersect if [schmall:293:308]
b1 −b
c −c
= 1
m−m1
n−n1
Intersection of two planes
Let the two planes be given by the equations [drmath] [osgood:477:502]
P 1 = A1 x B1 y C 1 zD 1 = 0
P 2 = A2 x B 2 y C 2 z D2 = 0
The line of intersection is
x −x 1
y −y 1 z −z 1
=
=
a
b
c
where
∣
a=
∣
∣
B 1 C1
B 2 C2
b=
∣
∣ ∣
C 1 A1
C 2 A2
c=
A1 B1
A2 B 2
 = a 2b 2c 2
∣
 x1 = b
∣ ∣
∣
D1 C 1
D B1
−c 1
,
D2 C 2
D 2 B2
∣
 z1 = a
 y1 = c
∣ ∣
D 1 B1
D
−b 1
D 2 B2
D2
∣
∣ ∣
∣
D 1 A1
D C1
−a 1
,
D 2 A2
D2 C 2
∣
A1
A2
If a = b = c = 0 , then the planes are parallel.
Intersection of three planes
The equations of the three planes are
A1 x B 1 y C 1 zD 1 = 0
A2 x B 2 y C 2 z D 2 = 0
A3 x B 3 y C 3 z D 3 = 0
and one must solve these simultaneous equations for the point x , y ,z . [salmon:21:46]
Intersection of four planes
For the four planes
A1 x B 1 y C 1 z D 1 = 0
A2 x B 2 y C 2 z D 2 = 0
A3 x B 3 y C 3 z D 3 = 0
A4 x B 4 y C 4 zD 4 = 0
to meet in a point, we must have [salmon:21:46]
∣
A1
A2
A3
A4
B1
B2
B3
B4
C1
C2
C3
C4
∣
D1
D2
=0
D3
D4
Intersection of a line and plane
Let r = rl + tnl be the equation of a line where rl is a point on the line and nl is a vector of the
direction cosines of the line. Let np·(r - rp) = 0 be the equation of a plane where np is the vector of
the direction cosines of the normal to the plane and rp is a point in the plane. Then the intersection
point is ri given by
r i = r l G n l
where
G=
n p⋅r p −r l 
nl⋅np
G won't exist if the plane is parallel to the line (i.e., the plane's normal is perpendicular to the line).
Another formulation [wilson:107:134]: let the equation of the plane be
r⋅n = 
where  is a constant and let us have a line that passes through the point r0 and is parallel to the
vector A; the equation is
A×r −r 0  = 0
Then the point of intersection is given by
 A×r 0×n A
A⋅n
This is simpler if n and A are unit vectors



 A×r
0 × n A
Angle between two lines in space
Suppose line L1 is given by r1 + sv1 and L2 is given by r2 + tv2. Then the angle between them (if
necessary, they must be translated so as to intersect) is
cos =
v 1⋅v 2
v1 v 2
If the two lines are characterized by vectors of their direction cosines d1 = (1, 1, 1) and
d2 = (2, 2, 2), then we have
cos = d 1⋅d 2
Angle between a line and a plane
Given a plane Ax By CzD = 0 and the line
x −x 0
y −y 0 z−z 0
=
=
a
b
c
The angle  between the line and the plane is the complement of the angle between the line and
the normal to the plane:
sin =
aAbBcC
 A2B 2C2  a2 b2c 2
Angle between two planes
Given two planes with normals n1 and n2. The angle  between these two planes is given by
cos =
n1⋅n 2
n1 n2
Parallel planes
If we have two planes
A1 x B1 y C1 zD 1 = 0
A2 x B2 y C2 zD 2 = 0
the condition that the planes are parallel is
A1 B 1 C 1
=
=
A2 B 2 C 2
If you convert to Hessian normal form, you can see that parallel planes will have the same direction
cosines in their equations. Equivalently, the dot product of their unit normal vectors is unity
n1⋅n2 = 1
Perpendicular planes
Two planes are perpendicular if the dot product of their unit normal vectors is zero:
n1⋅n2 = 0
Plane through a point parallel to two vectors
The plane parallel to a and b and through r0 is [coffin:232:261]
[ r −r 0 a b ] = 0
If we're given the direction cosines 1 , 1, 1  and  2 , 2, 2 of the two vectors then the equation is
∣
x −x 0
1
2
y −y 0
1
2
∣
z −z 0
1 = 0
2
Plane through a point parallel to another plane
The plane through r1 parallel to the plane of r2, r3 is [crc:356]
r = r 1s r 2t r 3
or
r −r 1 ⋅r 2 ×r 3 = 0
or
[r −r 1 r 2 r 3 ] = [r r 2 r 3 ]−[r 1 r 2 r 3 ] = 0
Plane through a line parallel to another line
Let the lines have direction cosines i = (i, i, i) and pass through ri = (xi, yi, zi) for i = 1, 2. The
equations of the planes containing one line and parallel to the other line are [salmon:31:56]
x −x i 1 2− 2 1   y −y i 1 2−2 1 
 z −z i 1 2−2 1  = 0 for i = 1, 2
or, in vector form,
r −r i ⋅ 1× 2 = 0
for i = 1, 2
The perpendicular distance between these two planes is
r 1−r 2 ⋅ 1× 2
sin
where  is the angle between the lines
sin =  1− 1⋅ 2
Plane through a given point and line
From [schmall:296:311]. Let the point be (x2, y2, z2) and the plane be given by
x −x 1
y −y 1 z −z 1
=
=
A1
B1
C1
The required plane is Ax By CzD = 0. Then, since the required plane must pass through the
given point and through the given line, we have the four equations
Ax By CzD = 0
Ax 2By 2Cz 2 = D
AA 1BB 1 CC 1 = 0
Ax 1By 1Cz 1 = D
Eliminate A, B, C, and D from these equations to get the equation of the required plane.
Plane through a point and normal to a vector
The plane normal to n and passing through r0 is
n⋅r −r 0  = 0
Line perpendicular to a plane containing three points
Suppose we have three noncollinear points r1, r2, and r3. The equation of the line perpendicular to
the plane containing these points is [crc:355]
r = r 1×r 2r 2×r 3 r 3×r 1
Line perpendicular to two lines
If we have two lines with direction cosines 1 = (1, 1, 1) and 2 = (2, 2, 2) and we want the
direction cosines  = (, , ) of the line perpendicular to both, we have [salmon:8:33]
=
 1× 2
 1× 2
=
sin 
 1− 1⋅ 2
where  is the angle between the two lines. Written out in components, this is
 sin =  1 2 −2 1
 sin = 1 2−2 1
sin  = 1  2−2 1
Distance of a plane to the origin
Suppose we have three noncollinear points r1, r2, and r3. The distance of the plane containing
these three points from the origin is [crc:355]
d=
r 1⋅r 2 ×r 3
r 2−r 1 ×r 3−r 1 
Distance of a point to a line
Given the direction cosines  =  ,  ,  and a line through r1 in this direction. Then the distance d
from a point r2 to the line is [drmath]
d = ×r 2−r 1 ⋅×r 2−r 1 
Distance of a point from a plane
n
Q
r
θ
D
P
R
Let Q = (x0, y0, z0) and n be a normal to the plane. The plane has the equation ax by czd = 0
and it does not contain the point Q. Now r = x −x 0 , y −y 0 , z −z 0 . Then
∣n⋅r∣ ∣ax 0 by 0cz 0d∣
D = r ∣cos∣ =
=
∣n∣
 a 2b 2c 2
If D is positive, the point and the origin are on opposite sides of the plane.
Distance between two non-intersecting lines
Let the direction cosines of the two lines be  1 = 1, 1, 1 and  = 2,  2, 2  and let the lines
pass through the points (x1, y1, z1) and (x2, y2, z2), respectively. The shortest distance between the
lines is
d =  x 1−x 2  y 1−y 2 z 1−z 2 
where



±1
=
=
=
1 2−1 2 1 2−2 1 1 2−1  2 sin
where  =  ,  ,  are the direction cosines of the shortest line connecting the two lines and  is
the angle between the two lines. We have
=
 1× 2
 1× 2
=
sin 
 1− 1⋅ 2
where  is the angle between the lines. In components, this is
 sin =  1 2 −2 1
 sin = 1 2−2 1
sin  = 1  2−2 1
The distance is [drmath]
±
∣
x 2−x 1
1
2
∣
2
∣
y 2−y 1 z 2−z 1
1
1
2
2
2
2
∣∣ ∣∣ ∣
1 1

 1
2 2
2
1
 1
 1
2
 2 2
The lines intersect iff the numerator is zero.
Circle in space
[http://mathforum.org/library/drmath/view/63755.html]
A parametric equation for a circle in space can be found from:
n = unit normal vector for the plane containing the circle
C = circle center
r = circle radius
u = unit vector from C toward a point on the circle
 u
v = n×
t = parameter
Then a point P is on the circle if
P = Cr  u cos t v sin t 
Vectors
First degree vector equation
[wilson:90:117] An equation of the first degree in an unknown vector r has each term as a vector
quantity containing the unknown vector not more than once. An example is
a×r b⋅c r  r −F = 0
The equation may be solved for r by dotting it successively with three noncoplanar vectors to get
three scalar equations. If the equation is of the form
Aa⋅r Bb⋅r C c⋅r = D
then one can dot multiply by A', B', and C' where the prime denotes the reciprocal basis vectors to
(A, B, C) (of course, they must be noncoplanar). Then one will get
r = D⋅A' a ' D⋅B' b ' D⋅C ' c '
which is the solution.
Other terms in the equation may be of the form  r or E ×r . These can be expressed as
[wilson:91:118]
 r =  a ' a⋅r b ' b⋅r c ' c⋅r
E ×r = E×a ' a⋅r E ×b ' b⋅r E ×c ' c⋅r
By such means, one will be able to reduce the equation to the form
L a⋅r M b⋅r N c⋅r = K
and this can solved as above using the reciprocal basis (see page 73) as
r = K⋅L ' a ' K⋅M ' b 'K⋅N ' c '
Invariance
Vectors that represent physical quantities need to be invariant under common coordinate
transformations. You can mentally picture the vector as an arrow (either as a free vector or
attached to a certain point, such as in a torque). Thus, for example, imagine you're applying a
torque to a lug nut with a socket wrench. A second person must measure the same force and lever
arm and calculate the same torque you're applying, regardless from where he views you.
A way to view this is that the torque vector has an "existence" independent of the coordinate
system that has been put on the space. Thus, we would only want to use mathematical constructs
and models that have the same behavior. The fact that the torque is independent of the observer
can only be found by experimentation, not deduced from first principles.
This invariance applies to the usual Euclidean transformations of everyday experience.
There are other cases where you can envision the vector being attached to the space. An example
would be making measurements of an object by photographing it. The position vectors you
measure will be affected by distortions from the imaging system. If you used a photograph to
measure the lever arm of the torque above, the perspective transformation (and distortion effects)
would be inherent in the physical measurement. Clearly, the calculated torque is not an invariant
measured value in this case -- we assume it is a physical invariant, but our measurement methods
include some distortion.
Spheres
The general equation of a sphere is [drmath]
x 2 y 2 z 22d x 2 e y 2 f z m = 0
The center is −d ,−e ,−f  and the radius is r =  d e f −m .
2
2
2
The equation of a sphere at the origin with radius R is [coffin:232:261]
r⋅r = R 2
The equation of a sphere with center at r1 and radius R is [crc:357]
r −r 1⋅r −r 1  = R 2
The equation of a sphere having points r1 and r2 as the endpoints of a diameter is [crc:357]
r −r 1⋅ r −r 2  = 0
Four point form [drmath]
∣
x 2 y 2z 2
2
2
2
x 1 y 1z 1
2
2
2
x 2 y 2z 2
x 23 y 23z 23
x 24y 24z 24
x
x1
x2
x3
x4
y
y1
y2
y3
y4
z
z1
z2
z3
z4
∣
1
1
1 =0
1
1
For the plane r·r1 = s to be tangent to the sphere (r - r2)·(r - r2) = R2, we must have [crc:357]
s −r 1⋅r 2⋅s −r 1⋅r 2=r 21 R 2
The equation of the tangent plane at r2 on the surface of sphere (r - r1)·(r - r1) = R2 is [crc:357]
r −r 2⋅r 2 −r 1  = 0
The area of the surface of a sphere of radius r illuminated by a point source a distance a away from
the surface of the sphere is [hawkes:491:210]
2 a r 2
ar
If a hole of length L is drilled through a sphere, the remaining volume of material is independent of
the radius of the sphere [old puzzle].
Miscellaneous
Center of mass
The vector r to the center of masses mi at the points ri is
r=
 mi r i
 mi
Projections
Isometric
Ref. http://en.wikipedia.org/wiki/Isometric_projection
The following is the isometric transformation for transforming a point r in the first octant to a two
dimensional point (bx, by) in the xy plane:
[ ] [ ][
bx
by
0
=
][
][ ]
1 0 0 1
0
0
cos  0 −sin  x
0 1 0 0 cos sin
0
1
0
y
0 0 0 0 −sin cos sin 0 cos z
=
[ ][
][ ]
1 0 0
3
0
− 3 x
1 
1
2
1
0 1 0
y
6

0 0 0
 2 − 2  2 z
1


−1
= sin tan
≈ 35.26° and  = . The leading matrix of 1's and 0's is an
6
4
2
orthographic transformation to the xy plane.
−1
where  = tan


The transformation matrix is thus
[ ]
1
2
1
6
0
0

2
3
0
−1
2
−1
6
0
The determinant is zero. The transformation equations are
x−z
2
x 2yz
by =
6
bx =
Projection angles
Z

z

r

y
x
Y

X
In the figure, the vector r has the spherical coordinates (r, , ). It also has the angles  and  that
are the projection angles onto the xz and yz planes, respectively.
We have
2
2
2
r =  x y z
x = r sin cos
y = r sin sin
z = r cos 
x = z tan 
y = z tan 
From
x = r sin cos = z tan = r cos tan 
y = r sin sin = z tan  = r cos tan 
we get
tan  = tan cos 
tan  = tan  sin
(1)
with inverses
tan
tan
(2)
2
2
2
tan  = tan tan 
These relationships are useful in the shop to document how something is constructed. You can
use sine sticks (see below) to measure the angles of a feature from two orthogonal directions and
tan  =
reconstruct the direction in space. This also can be done by taking a photograph from the two
orthogonal directions and measuring the angles from the pictures. It's best to back away from the
object and use a telephoto lens to minimize added distortions.
Sine sticks
A simple-to-make device can be used to measure projections angles and, thus, measure angles in
space.
A
Step 1: drill holes
L
Step 2: rip longitudinally along AB
B
Take a board and drill two holes through it a distance L apart. Then rip the board in half
longitudinally to get two identical pieces. Join the two boards with a snug-fitting bolt or rivet through
the holes so that the two boards can pivot about the hole center. Tighten the bolt enough to let the
sticks' angle be adjusted but held in place with normal handling.
To use this device, adjust the edges on the two boards to coincide with an angle.
α
θ = angle to be
measured
Measure the distance  and calculate the angle  by
 = 2 sin−1

2L
On a practical note, a 12 inch or larger metal folding rule is convenient to make such an angle
measuring device (one such rule was the Lufkin blacksmith's rule). Just superimpose the two legs,
clamp, and drill a small hole through both legs -- these holes will automatically be at the same
distance from the pivot. You can then carefully calibrate the device to a right angle to determine
the distance L. In use, dividers can be used to set or measure the distance . The author has
seen such a rule from the early 1900's that had such marks and a table to help measure angles.
Some reflection will show you this angle measuring device can be surprisingly accurate. For an
angle of 60°, if you made a measurement error of 0.02 inches for an L of 10 inches, you'll see that
this affects the angle by 0.13°. Thus, ordinary shop work can be done to a tenth of a degree
without undue effort.
For closer work, these sine sticks can be made from metal such as some 0.25" by 0.5" aluminum
rectangular bar stock. Drill and ream the holes, then press in (or use Loctite) some pins with center
marks located on a lathe. Using trammels, you should be able to measure the separation's  to
0.005" or less. The above 60° angle will be in error by 0.033°, which is nearly 3 times better than
the usual machinist's vernier protractor can read (which is typically 5 minutes of arc). Even more
accurate measurements could be made by using a sine bar and using the sine sticks as an angle
transfer device.
Rotations
See "Rotation matrix" at http://en.wikipedia.org/wiki/Rotation_representation_(mathematics) and
http://en.wikipedia.org/wiki/Rotation_matrix. Matrix representations of SO(n) have a determinant of
+1 and are orthogonal matrices, meaning their transpose is equal to the inverse. For n > 2, the
group is non-Abelian (i.e., rotation matrices do not commute). This is easy to see from the
infinitesimal generators; e.g., calculate the commutator for y ∂ x −x ∂y and z ∂ x −x ∂z.
The following matrices are SO(3) representations and represent rotations about the x, y, and z
axes in 3-space:
[
[
[
1
0
0
R x = 0 cos −sin 
0 sin cos
R y =
cos  0 −sin 
0
1
0
sin 0 cos
cos  −sin 0
R z = sin  cos  0
0
0
1
]
]
]

Note that if  = /2, the Rx takes j into k , Ry takes i into k , and Rz takes i into k.
Other rotation matrices can be gotten from these three using matrix multiplication. Thus,
R x R y R z  represents a rotation whose yaw, pitch, and roll are , , and , respectively.
Similarly, R z R x R z  represents a rotation whose Euler angles are , , and .
For using Euclidean transformations, it is handy to use homogeneous coordinates: the general
position vector is (x, y, z, 1). The transformation matrices are then
[
1
T a , b ,c  = translation = 0
0
0
[
[
[
0
1
0
0
0
0
1
0
a
b
c
1
]
]
]
]
1
0
0
0
0
cos

−sin
0
R x =
0 sin cos  0
0
0
0
1
cos
0
R y =
sin
0
0 −sin
1
0
0 cos
0
0
cos −sin
sin cos
Rz =
0
0
0
0
0
0
1
0
0
0
0
1
0
0
0
1
One can also write the rotation matrix as the direction cosine matrix. Suppose we have three unit
 as the basis of the rotated system
vectors u , v , w
[
ux vx w x
R = uy vy w y
u z vz w z
]
where each element is the cosine of the angle between a rotated unit basis vector and one of the
reference axes. R is a real, orthogonal matrix with unity determinant and eigenvalues
1, cosi sin , cos −i sin 
The eigenvalue of 1 corresponds to the rotation axis, as it is the only vector unchanged by the
rotation. R has three degrees of freedom; they are constrained by the relations
∣u ∣ = ∣v ∣ = 1
 v = 0
u⋅
 = u×
 v
w
A rotation matrix preserves distances between points. Let r be an arbitrary vector; then rTr is the
square of its magnitude. If Rr is a rotated vector corresponding to the rotation matrix R, we have
(Rr)T(Rr) = (rTRT)(Rr) = rT(RTR)r = rTr, which again is the square of r's magnitude. RTR is the
identity matrix because R is orthogonal.
Rotating about a given axis
Given a unit vector u = x , y , z , the matrix for a rotation by an angle  about an axis in the
direction of u is (http://en.wikipedia.org/wiki/Rotation_matrix#Finding_the_rotation_matrix)
[
2
2
x 1−x cos 
x y  1−cos −z sin x z 1−cos  y sin
R = x y 1−cos z sin 
y 21−y 2cos 
y z 1−cos −x sin
x z 1−cos −y sin  y z 1−cos x sin
z 2 1−z 2cos 
]
In a normal right handed Cartesian coordinate system, this rotation will be counterclockwise for an
observer placed so that the axis u goes in the observer's direction (i.e., the tip of the arrow that
represents u pokes the observer in the eye).
Rodrigues' formula can be used instead:
R = P I −P cos Q sin
where
[
2
x
xy
P= x y y 2
x z yz
]
[ ] [
xz
1 0 0
0 −z y
T
=
u
u
,
I=
,
Q
=
yz
0 1 0
z
0 −x
0 0 1
−y x
0
z2
]
Q is the skew-symmetric representation of the cross product, P is the projection onto the axis of
rotation, and I - P is the projection onto the plane orthogonal to the axis.
Direction cosine matrix to Euler angles
Given a 3x3 rotation matrix R, one can calculate the zxz Euler angles , ,  (rotation around z, x,
then z) as:
 = atan2R 31 ,R 32
−1
 = cos R 33  must be in [ 0, )
 = −atan2R 13 , R 23
If R33 = 0,  and  should be calculated from R11, R12 instead. atan2 is the usual atan2(y, x) found
in programming languages (note the y component is the first argument).
Direction cosine matrix to Euler axis and angle
If the Euler angle  is not a multiple of , then the Euler axis u = (x, y, z) and angle  are
trace R −1
2
R 32−R 23
x=
2sin 
R 13−R31
y=
2 sin
R 21−R 12
z=
2 sin
cos =
This is the eigenvector associated with the eigenvalue of 1.
Rotating lines in space
(Motivation: http://www.mathforum.org/library/drmath/view/56462.html.)
Suppose you have a notebook (see figure below). Open it up and draw a line L1 from a point O on
the spine to anywhere on the right page; call the angle between the spine and the dashed green
line a. On the left page, draw a line L2 from O to anywhere; call the angle between the spine and
the left dotted line b. Now, open the notebook so that the angle between the two pages is c. Then
the angle  in space between the two drawn lines L1 and L2 is
cos = cos a cosbsina sinb cosc
Z
c
L2

L1
b
a
v
u
X
O
-Y
Useful sub-cases are:
c = 0°:  = a−b
c = 90°: cos = cos a cos b
c = 180°:  = ab
Derivation: suppose that the dashed and dotted lines both pass through the origin O as shown and
a customary right-hand Cartesian coordinate system is put where the lines intersect, as shown. Let
u be a unit vector pointing in the dashed line's direction and v be a unit vector pointing in the dotted
line's direction. Then the cosine of the angle  is
cos = u⋅v
Since u lies in the xy plane, its direction cosines are easy to write down: sin a , cosa , 0.
To get v's direction cosines, let's start with c = 0. Then v's direction cosines are sin b , cosb ,0.
Rotate the coordinate system about the y axis counterclockwise by an angle c. Clearly, only the x
and z components will change. The y-axis rotation matrix is
R y =
[
cos  0 −sin 
0
1
0
sin 0 cos
]
Here are the direction cosines of the rotated direction cosine vector of the dotted line L2 up out of
the xy plane by the angle c:
[
]  
cos c 0 −sinc sinb
sin b cosc
=
0
1
0
cosb
cos b
sinc 0 cos c
0
sin b sinc

Now we calculate the angle :
cos = sina , cosa , 0⋅sinb cosc , cosb , sin b sinc 
or
cos = sina sinb cos ccos a cos b
Helix
Ref. Weisstein, Eric W. "Helix." From MathWorld--A Wolfram Web Resource.
http://mathworld.wolfram.com/Helix.html
A helix is a curve in space on a right circular cylinder that advances at a uniform rate as the angle
around the cylinder's axis changes. A screw thread is the prototypical example. The parametric
equations of a helix are
x = r cos 
y = r sin
z = p
r is the radius of the cylinder and 2p is the pitch of the helix. The tangent line to the helix makes a
fixed angle with an arbitrary line. The helix is the shortest path between two points on a cylinder
(cut the cylinder, roll it out flat, and draw a line between the two points).
Formulas:
 = r 2p2
Arc length =   
r
Curvature =  =

p
Torsion =  =


r
= = constant
 p
A curve is a helix iff the ratio of curvature to torsion is constant.
Solid angles
For a solid angle  in three dimensions, we have the following picture:
C

R
A small circle C is on the surface of the sphere of radius R. The solid angle is then
=
A
R2
where A is the area of the spherical cap with the circle C as its border. A solid angle is a
dimensionless number; it is also given a name, the steradian. If you make the circle C a great
circle of the sphere, you can see that the solid angle is 2 and it's 4 for the whole sphere. If you
make the area of the cap equal to R2, then you have a unit solid angle. You can convert steradians
to square degrees by multiplying by 180 /2, analogous to converting from radians to degrees by
multiplying radians by 180/,
Suppose I have an area A that subtends a solid angle  from point S as shown in the figure below.
If I rotate the area A about the axis PQ by an angle , the solid angle subtended by the circle from
point S is reduced to cos .

P
Area A
S
C


Q
Figure 3
If  is the angle subtended by the radius of the circle and the plane of the circle is perpendicular to
SC in Figure 3, the solid angle subtended by the cone with vertex at S and intersecting the circle's
circumference is
 = 21−cos 
Layout
This section contains various methods of geometrical layout using simple tools.
Cutting pipe ends for welding
Suppose cylindrical pipe 1 of radius R runs along the y axis and another cylindrical pipe 2 of radius
r where r ≤R runs along the x axis. Rotate pipe 2 by an angle  about the z axis (the positive angle
is counterclockwise when looking at the xy plane from the +z direction) and then translate the pipe
in the +z direction by a distance a. Since the generator for the rotation is x ∂y − y ∂x and the
translation is ∂z , these transformations commute.
You can lay out a pattern on a sheet of paper and use it as a template to cut the end of the smaller
pipe. The formula for the y distance to trim off given the angle  around the circumference of the
smaller pipe is
y  = sec    R −r sin a r cos  sin  
2
The angle =0 is in the plane containing the pipes' axes.
2
To derive this, start with the parametric equations for one cylinder that lies on the y axis:
f u ,  = r cos  , u , r sin 
Start with the other cylinder lying along the x axis:
(3)
g  v ,  = v , R cos  , R sin 
Use a matrix to rotate this vector an angle alpha around the z axis:

cos  −sin 
M = sin  cos 
0
0

0
0
1
This leaves the cylinder in the xy plane. Then translate it in the +z direction.
Equate the components f and g, then solve them for θ, u, and v in terms of  and substitute into (3)
to get a one-parameter vector for the curve of the intersection. Then use the inverse of M to rotate
the curve back to it around the x axis and translate back down to the xy plane. This will yield the
above formula.
The site http://www.harderwoods.com/pipetemplate.php provides an on-line solution via this
method.
To make a template to trace the hole on the larger pipe, start with the parametric vector describing
the curve:
 R −  r sin a  r cos  sin 
2
2
x=
cos 
y = r cos 
z = r sin −a
Looking down the y axis (the axis of the larger pipe), you'll see
B
R
z

A
x
O
−1
The point B is on the curve of intersection. The angle  = tan
−1
s = R . Then when we plot the points (R tan
be traced in the plane.
z
. The arc length of the arc AB is
x
z 
, y ) as  goes from 0 to 2 , the cutout will
x 
Dividing a line equally
The quickest method is to lay a rule at an angle to the line, and mark off points along a line at n
major divisions. Then draw parallel lines from the marked points to intersect the original line (this is
most easily done with a drafting machine or two triangles). It the figure below, AB is the line to be
divided equally into 4 parts. Suppose we set a rule along line BC that makes BC exactly 4 units.
Then we mark the points 1, 2, 3, and 4. Draw a line from point 4 (i.e., C) to A. Then draw lines
through 1, 2, and 3 parallel to AC. The intersections divide line AB up as desired.
4
C
3
2
1
A
B
You can also calculate AB/n and use your rule to mark successive points on the line AB. But this
method can be tedious and subject to reading errors because the required points are e.g. not
integers. The above method, used for thousands of years, avoids the tedium and reading mistakes
(you can also use a compass or dividers to mark of equal lengths on line AC).
Because calculators are common, you can calculate AB/n and set the dividers to this value, then
step off the intermediate points. The first method can be more accurate, as there's no cumulative
error due to setting the dividers slightly wrong. However, you can eliminate the cumulative error if
you first step off the divider along AB and ensure it comes out the proper length; if not, make small
adjustments and repeat. This is most easily done with machinist dividers that are adjusted with a
fine-thread screw or dividers made for drafting. If the error at the end of stepping off n points is ,
then the divider setting error is /n. With a little care, you can divide AB up into n equal divisions
with no sensible error.
Geometric mean
To construct the geometric mean of two numbers a and b, construct the following figure
P
a
A
b
B
C
The length of BP is the geometric mean of a and b or  ab.
Proof: angle APC is a right angle; thus, angles APB and PCB are the same. The tangents of APB
and PBC give a/PB = PB/b and the relation follows.
Golden ratio
[marks:2-14]
C
D
r
r=
AB
2
θ
A
E
B
To divide a line AB into two segments AE and EB that have the golden ratio
AE
1 5
=
= 1.6180
EB
2
construct BC perpendicular to the line AB at B such that the perpendicular's length r is half of the
line AB's length. Draw AC and scribe arc DB with center at C and radius r. Then scribe arc DE
with center at A. E is the required point that divides AB into the desired ratio.
Proof: Let AB = a ; then AC = r  5 and AD = AE = AC−r = r   5−1. Then
EB = a−AE = 2 r −r   5−1 = r 3−  5
Hence
AE
r   5−1 1 5
=
=
EB
2
r 3− 5
Inscribed/circumscribed circle, etc.
To construct the circumscribed circle of a given triangle, the center of the circle is the intersection of
the perpendicular bisectors of the sides. For the inscribed circle, the center is intersection of the
bisectors of the interior angles.
To inscribe a hexagon in a circle, draw a diameter. Then scribe the radius from the intersections of
the diameter and the circle; the intersection points are on the hexagon.
Line segment perpendicular bisector
Given line segment AB, construct the perpendicular bisector by scribing equal arcs from the
endpoints of the segment. Set dividers to 3/4 or more of the line segment length. The intersections
of the two equal circular arcs lie on the perpendicular bisector.
r
r
B
A
r
r
The method works because the diagonals of a rhombus are perpendicular bisectors of each other.
Perpendicular to a point interior to a line segment
Given line AB and point C to which a perpendicular to AB must pass. Using dividers, construct arc
DE with center at C:
F
A
D
C
E
B
Then construct equal radius arcs from centers D and E that pass through point F. Line CF is
perpendicular to AB.
Perpendicular at end of line segment
Given line segment AB. To construct a perpendicular at B, set dividers to a reasonable size and
scribe arc CD with center at B. Locate point E on arc CD approximately 45° above the line AB.
Scribe the dashed circle FBG with radius EB. Construct line FG and extend it to intersect the
dashed circle at G. The line BG is perpendicular to AB.
G
C
E
A
B
F
D
The principle is based on the fact that if any two chords FB and GB in a circle meet along a
diameter EF, then angle FBG is a right angle.
Bisecting an angle
To bisect angle ABC, construct arc DE with center B. Construct arcs DF and EF that have the
same radius. BF is the angle's bisector.
A
D
F
B
E
C
Inaccessible point
A point P is given along with two lines AA' and BB' that intersect at an inaccessible point Q.
Construct the line PQ. The method is to use similar triangles:
A
C
P
D
P'
A'
E'
E
B'
B
D'
C'
Draw any line CC' and construct DD' parallel to it. Draw CP and construct DP' parallel to it. Draw
PE and construct P'E' parallel to it. Now you have the point P', so construct the desired line
through PP'.
Laying out right angles
Two triangles are commonly used for right angle layout: the isosceles and the 3:4:5 triangle.
√2
1
5
π/4
1
4

3
These are often used in carpentry, yard work, and surveying. The advantage is that all that's
needed is a tape measure. The isosceles triangle method can be extended to measuring the
diagonals of a square -- when they are equal, then all four corners are right angles. This method
doesn't even need a tape measure -- mark one diagonal on a stick and compare it to the other
diagonal.
The 3-4-5 triangle is also useful with dividers, as it is rare one doesn't have a rule in the shop. Be
−1 4
= 0.9273 radians = 53.13°.
mindful that angle  isn't any "common" angle -- it's tan
3
Copying an existing angle
Suppose angle ABC is given and it is desired to copy this angle. The new vertex is B' and one side
of the angle is to fall on line B'D'. On the ABC angle, draw an arc ED and, without changing the
divider settings, draw the arc E'D' from the new vertex B'. On the ABC angle, set the dividers to the
distance DE and scribe D'E' on the new drawing. Set the dividers to center D and scribe the arc
EF; scribe the corresponding arc E'F'. Then draw the line B'E'.
A
E
E'
F
F'
B
D
B'
C
D'
Dividing an angle
To divide an angle ABC into n equal parts, draw arc DE with center at B, then set the dividers to an
estimate of the arc length DE/n. Step off n points on the arc DE. Iterate until the last point falls
exactly on point D or E. The diagram shows the case when n = 3.
A
E
B

r
D
C
You cannot draw a line from DE and divide that into equal segments to equally divide the angle.
However, if you have a rule and calculator handy, you can measure the line segment DE and r,
then calculate the angle  by the cosine law. Then the divider setting is
2r sin

2n
That using a straight line doesn't work is shown by trying to divide a right angle into three equal
angles. The following sketch shows the problem:
a
1
a
1/3
a
θ
1/3
2/3
1
1/ 3
= 26.56°, which doesn't lead to the desired
2 /3
90 °
=2sin15 ° = 0.5176.
angle of 30°. However, the formula above gives a divider setting of 2 sin
6
2 sin15 °/  2
This gives the correct angle of tan−1
= 30 °.
1−2 sin15 °/  2
−1
Since a =  2/3 = 0.4714, we see that  = tan
Laying out angles
It can be useful to remember the central angle theorem of Euclidean geometry:
A
2θ
B
θ
C
The central angle 2 in red subtends an arc on the circle. The blue dashed angle with vertex on
the circle that subtends the same arc on the circle has an angle of . The theorem is true for any
point C on the circle as long as C is not between A and B. A corollary is Thales' theorem: an angle
whose vertex is on the circle and that subtends a diameter is a right angle.
Using tangents
Most folks are familiar with protractors, but because they are typically small, they can't be used for
accurate angle layout. Drafting triangles are better, as they are typically made to close tolerances
and a well-made drafting machine can be good too. My favorite method, however, is to use
trigonometric relationships, typically the tangent. Here's an example of laying out an angle on the
table top of a table saw (this is used e.g. to set a guide when one wants to cut a cove).
saw blade
C
y

x
A
B
You can measure distance AB on the table. Then to get an angle  with respect to AB, measure up
a distance y = x tan  on BC. For accurate work, this requires that ABC is a right angle; if it isn't,
you can measure it and still make the layout, although you'll have to use the cosine law.
Here's an estimate of the angle error:
d=

−y dx x dy 
x2
where  =
1
2

y
1
x
For my table saw, the distance x is 28.2". For an angle of 30°, we have y = 16.28". If we assume
1.333
. Reducing the dx and
dx = dy = 0.1 inches , then d =
2 −1.6282.82 = 2 mrad or 0.11 °
28.2
dy uncertainties will reduce the d uncertainty proportionately.
An alternative technique is to use a carpenter's square. Since these are annoyingly graduated in
fractional inches in the US, one needs to use a table or a calculator.
Another technique for laying out angles is to use dividers and a rule for construction, as in the
following examples.
60°
The radius of a circle is used to layout an angle of /3 = 60°:
B
60°
O
A
Set the dividers to OA, the radius of the circle, then scribe the arc through B with center A. OB is at
a 60° angle to OA.
45°
Bisect a right angle. A quick template can be made from paper with square edges: fold one corner
over so that its edge lays along the an opposite edge. The fold will be at 45°.
30°, 15°
Make a 60° angle and bisect it once or twice.
Other special angles
Here's an example of making a 40° angle. Other special angles can be made through
combinations of the techniques shown here. In general, however, the method of using the tangent
above will probably be found to be the quickest as long as one has a calculator or table of tangents
handy. Sometimes these special methods can be used when no tables or calculator are handy, so
they're useful to have in your back pocket.
Make a 45° angle by bisecting a right angle and make a 30° angle by bisecting a 60° angle. Now
divide the 15° angle into three equal angles and the needed 40° angle will be 5° less than the 45°
angle.
Triangles
Inscribed circle
The bisectors of the internal angles of a triangle intersect at the center of the inscribed circle.
Circumscribed circle
The perpendicular bisectors of the sides of a triangle intersect at the center of the circumscribed
circle.
Hexagon
To layout the inscribed hexagon within a circle, lay out the radius along the circumference. For the
circumscribed hexagon,
D
B
E
C
O
A
r
lay out radius r as AB and draw its perpendicular bisector OE. Construct tangent CD that intersects
line OB. Then lay out radius CD around the dashed circle as in the inscribed circle case.
Octagon
From each corner of a square, scribe the dashed circles with a radius of one-half of the square's
diagonal. Then the required octagon is at the indicated intersections of these arcs with the square.
Ellipse
In the following section on ellipses, 2 a is the major diameter of the ellipse and 2 b is the minor
diameter.
The area of an ellipse is a b and the perimeter is 4a E where E is the complete elliptic integral of
2
2
the second kind with k =  a −b /a . Ramanujan's rather amazing 1914 approximation for the
perimeter is
[
p ≈  ab 1

10  4−
]
, where =3
Here's a graph showing how good this approximation is for a = 1:
2
 
a−b
ab
formula
. The noise above b = 0.8 is likely due to complete loss of
exact
significance in the subtraction.
The curve plotted is 1−
Piece of string and two foci
Set the dividers to a and scribe the arc from A to intersect the major diameter at points B and C.
Points B and C are the foci. The foci have the property that any beam of light BDC emitted from
one foci will arrive at the other foci, assuming the interior of the ellipse is perfectly reflecting. The
2
2
abscissas of the foci are OC =  a −b .
A
D
a
b
E
B
O
C
a
The ellipse has the property that any point D on the ellipse has a constant sum of distances from
the foci (i.e., BD + CD = 2a).
To construct the ellipse, push two pins or nails in at the foci. Then tie a string such that the loop
causes the pencil point, when stretching the loop taut, to mark at point A. Then you should be able
to trace the ellipse by keeping the pencil point taut in the string loop. The loop length needs to be
BD + DC + BC which is equal to
2
2
2ad  = 2a a −b .
While the method is exact, it can be difficult to keep the pencil perpendicular and the string
stretched a uniform amount. Thus, don't expect perfection. If you're using a wooden pencil, file a
small groove near the tip in the wood to help guide the string.
Calculated points
Since the equation of the ellipse is
x2 y2
 =1
a2 b2
the required points can be calculated and plotted. A polygonal approximation to the ellipse can be
gotten, good enough for many shop tasks, especially if a little smoothing is done with a file or
sander.
This method is relatively fast if a template is made -- and only a quarter of the ellipse's form has to
be generated, as the other three quarters can be gotten by reflection (turning the template over)
and rotation.
Using a marked stick
Using the following ellipse, we mark a stick or card with the dimensions a and b.
b
O
a
a
b
O
a
b
O
Once the stick is marked, one can lay out the ellipse by keeping the points marked a and b on the
horizontal and vertical axes, respectively, and marking points on the ellipse at the mark O.
By moving the stick, one can mark out different points on the ellipse at point O, as long as the
points marked a and b are in contact with the major and minor diameters.
If you're doing this on a surface that you can attach a framing square to, then instead of using
marks, you can drive nails through the stick at the a and b marks (file their tips off a bit so they're
not so sharp after driving them through the wood). Drill a hole at O large enough to hold a pencil
and you can draw the ellipse by keeping the nails against the square (the square's right angle is
used to mark the coordinate axes). Use double-sided tape or clamp the square to the work.
Approximate Ellipse with circles
[blandford:107]
y
A
B
BC = AB/3
C
D
a
O
E
x
b
Figure 4
In Figure 4, construct the ellipse's axes and mark off the semimajor axis a (OE) and semiminor axis
b (OB). Mark arc DAE with center at O and radius of a. Mark BC to be a length of 1/3 of AB.
In the following diagram, we'll only show the constructions in one quadrant, but the extensions to
the other quadrants is obvious. The dotted line shows the true ellipse.
F
D
G
a
O
b
J
H
Figure 5
In Figure 5, set the dividers to distance OC in Figure 4 and mark the arc FGH with center at D.
Without changing the dividers, mark the arc FDH with center at G. Mark point J from the
intersection of the line FG with the y axis.
A
K
a
O
b
J
Figure 6
In Figure 6, draw arc AK with center at J, blending in with the green arc. The dots show the desired
ellipse and the red and green arcs show the approximation.
Parabola
Given the vertex V and a point P on the parabola.
A
B
6
5
4
3
2
1
V
1
2
3
5
4
6
P
X
Assume the x axis extends from V to the right along VX. Draw line BP parallel to VX through P.
Then construct the line AV perpendicular to VX. Mark n equally-spaced points (n = 6 shown in
diagram) along VA with the nth point ending on BP. Mark n equally spaced points between the
intersection of BP and AV and point P. Draw horizontal lines through the points on VA and lines
from V to the points on BP. The points at the intersection of lines with the same number are on the
parabola.
Regular polygons
Let the polygon have n sides. The geometry is
B
D
R

C
r
O
R
A
Figure 7
The blue dashed outside circle is the circumscribed circle and the green dashed inside circle is the
inscribed circle. The central angle is  = 2 / n . The relationship between R and r is
r = R cos


= R cos
2
n
The easiest way to lay out the polygon is to scribe the circumscribed circle ADB, then step off
consecutive points with the dividers set to distance AB. Any setting error is cumulative, so a trial
run should be done to ensure you wind up back at the starting position. The divider setting is
AB = 2 R sin


= 2 R sin
2
n
You can avoid the cumulative error by calculating the divider setting for each point using the cosine
law and always having one leg of the divider at point A:
divider setting = R  21−cosi  where i =
2
i
Accuracy of this method, however, is less when  is near 180° because of loss of significance. In
addition, it is more tedious and, thus, more prone to setting errors. However, if n is even, then you
can scribe two points per setting from A and two more points from the point corresponding to A at
the other end of the circumscribed circle's diameter.
Another method which avoids the cumulative error but is also more tedious is to use the Cartesian
coordinates of each point. These are gotten through the usual polar conversions:
2
i
2
y i = R sin i = R sin
i
x i = R cos i = R cos
2
. Machinist's handbooks [mh71] give tables of the
i
coordinates, as toolmakers used to use them on jig boring machines.
where i = 0, 1, 2, ..., n -1 and i =
Arcs
Suppose it is necessary to lay out the circular arc ABC as shown in the figure, but the center of
circle is either inaccessible (e.g., inside a wall) or inconveniently long. Use Cartesian coordinates.
B
E
h
s = AC
y
C
D
A
x
In the diagram, suppose we know h and the radius r of the arc ABC. Then the coordinates (x, y) of
the point E on the arc with the origin at point D has the ordinate y
2
2
y =  r −x −r h
If you have the arc ABC and know the points A and B, but don't know the radius r of the circle, you
can calculate it from measuring s = AC and h = BD:
r=
s 24h2
8h
It should be evident that if r is large and h is small, small measurement errors can result in large
uncertainties in r.
A construction method to get the radius and the center of the arc is to construct the perpendicular
bisectors to two chords of the circle. If the center is not accessible, then the line BD that it is on
can be gotten by constructing the perpendicular bisector of the chord AC.
Tapered boxes
Suppose we want to make a symmetrical tapered box from some thin material. The box is as
follows:
C
D
A
B
In the vertical plane containing the points A, B, C, and D, we have the following view:

In the developed pattern for the box, the question is: given , what is the angle ?

O
cos=
This was derived by Jerry at Dr. Math.
1
 1sin2 
Useful tips
If you have a small angle measured in degrees such that the angle is less than about 5° (i.e,
roughly 1/10th of a radian), the sine and tangent of the angle can be calculated by dividing the
angle by 57.3 (i.e., 180/); this is suitable for hand calculations to roughly 0.1%. (This
approximation was well-known to slide rule makers for the ST scale and comes from the first term
in the Taylor series for the sine.)
Suppose you want this to approximation to be within 0.1% for hand calculations. Then the value
of x in radians for which this is true is the solution of the transcendental equation
sinx −x = 0.001 . Using the Taylor series, we get the approximate equation x 3 / 6 = 0.001
which gives us x = 0.18 radians.
For hand calculations, some useful rational approximations are:
 ≈ 22 /7 0.04%
 2 ≈ 99 /70  0.005%
 3 ≈ 97/ 56  0.005%
ln10 ≈ 175 /76 0.002%
loge ≈ 76/ 175 0.002%
log ≈ 87 /175 0.001%
The percentages in parentheses are the differences from the actual values.
10
1−0.01 . This makes it easy to use in a hand
7
calculation because in a multiplication by  2, you just shift the decimal point, divide by 7 to get the
result R, then subtract R/100 (shift the decimal point) from R to give the result.
The approximation to  2 can be written
This is a general technique in hand calculation because of the approximation from the generalized
binomial theorem:
1±a 1±b ≈ 1±a ±b if  ≪ 1 and  ≪ 1
For example, to calculate
106×84
37
we write
10010.06×8010.05 100×80
≈
10.0610.0510.075
40 1−0.075
40
Using the approximation again results in
200 10.060.050.075=200 1.185  = 237
We're low by 1.5%.
If you have to calculate the tangent of an angle near a multiple of /2, you can use the relation
∣tanx ∣ =
∣
1
∣

tan x− 
2

= cotx − 
2
∣
∣
This is useful for angles near /2 that have large tangents.
References
The books listed with a gray background were downloaded from Google books.
[albert]
Albert, A., Solid Analytic Geometry, McGraw-Hill, 1949.
[anderson]
H. Anderson, ed., Physics Vade Mecum, American Institute of Physics, 1981.
[as]
M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, Dover
Publications, 5th printing, 1968.
[bartsch]
Bartsch, H., Handbook of Mathematical Formulas, (translation of 9th German ed.),
Academic Press, 1974.
[blandford]
Blandford, P., The Master Handbook of Sheetmetalwork, TAB Books Inc., 1981
[coffin]
Coffin, J., Vector Analysis, Wiley, 1911.
[corral]
Corral, Michael, Vector Calculus, published by author on web, 2008,
http://www.mecmath.net/.
[crc]
Beyer, W., CRC Standard Mathematical Tables, 26th ed., CRC Press, 1981.
[drmath]
http://mathforum.org/dr.math/faq/formulas/faq.ag3.html#threeplanes Used with
permission. Dr. Math is a great web resource and I encourage you to visit the site
-- you'll find all kinds of useful stuff plus great volunteers to help you with math
questions.
[feynman]
R. Feynman, The Feynman Lectures on Physics, Addison-Wesley, 1963. In
particular, see chapter 22 in volume 1, as it discusses how logarithms are
calculated; the discussion ends with Euler's jewel.
[gr]
I. Gradshteyn and I. Ryzhik, Table of Integrals, Series, and Products, 4th ed.,
Academic Press, 1965.
[hawkes]
Hawkes, H., Luby, W., Touton, F., Solid Geometry, Ginn & Co., 1922.
[hudson]
R. Hudson, J. Lipka, H. Luther, and D. Peabody, A The Engineer's Manual, Wiley,
1917. While some of the engineering focus is obsolete, this is a superb book. In
the US, you can download it from Google books. If you're just interested in the
math stuff, a shorter version is R. Hudson and J. Lipka, A Manual of Mathematics,
Wiley, 1917.
[huntington] E. Huntington and L. Fischer, Mathematical Tables, which was taken from section
1 of Mark's Mechanical Engineers' Handbook, McGraw-Hill, 1916. This book has
excellent tables for doing hand calculations. You can download this as a book
from Google books (it's only 5 MB compared to the whole 100 MB edition of
Marks).
[lawrence]
J.D. Lawrence, A Catalog of Special Plane Curves, Dover, 1972, 0-486-60288-5.
[marks]
L. Marks and T. Baumeister, Standard Handbook for Mechanical Engineers, 7th
ed., McGraw-Hill, 1967. You can download the 1916 edition from Google books,
but be warned it's 100 MB.
[mh04]
E. Oberg, F. Jones, H. Horton, H. Ryffel, Machinery's Handbook, Industrial Press,
27th ed., 2004.
[mh71]
E. Oberg and F.D. Jones, Machinery's Handbook, Industrial Press, 19th ed., 1971.
[osgood]
Osgood, W. and Graustein, W., Plane and Solid Analytic Geometry, Macmillan,
1921.
[perry]
R. Perry and C. Chilton, Chemical Engineer's Handbook, 5th ed., McGraw-Hill,
1973.
[salmon]
Salmon, G., A treatise on the analytic geometry of three dimensions, 4th ed.,
Hodges, Figgis & Co., Dublin, 1882.
[schmall]
Schmall, C., A First Course in Analytical Geometry, 2nd ed., van Nostrand, 1921.
[snyder]
Snyder, V. and Sisam, C., Analytic Geometry of Space, Henry Holt and Co., 1914.
[wilson]
Wilson, E., Vector Analysis, Yale University Press, 1922. Based on the lectures of
Gibbs.