Geometry
(Many slides adapted from Octavia Camps and Amitabh
Varshney)
Goals
• Represent points, lines and triangles as
column vectors.
• Represent motion as matrices.
• Move geometric objects with matrix
multiplication.
• Refresh memory about geometry and
linear algebra
Vectors
v  ( x1 , x2 ,  , xn )
• Ordered set of
numbers: (1,2,3,4)
v
• Example: (x,y,z)
coordinates of pt in
If
space.

n
2
i 1
i
x
v  1, v is a unit vecto r
Vector Addition
v  w  ( x1 , x2 )  ( y1 , y2 )  ( x1  y1 , x2  y2 )
v
V+w
w
Scalar Product
av  a( x1 , x2 )  (ax1 , ax2 )
av
v
Inner (dot) Product
v

w
v.w  ( x1 , x2 ).( y1 , y2 )  x1 y1  x2 . y2
The inner product is a SCALAR!
v.w  ( x1 , x2 ).( y1 , y2 ) || v ||  || w || cos
v.w  0  v  w
Points
Using these facts, we can represent points.
Note:
(x,y,z) = x(1,0,0) + y(0,1,0) + z(0,0,1)
x = (x,y,z).(1,0,0)
z = (x,y,z).(0,0,1)
y = (x,y,z).(0,1,0)
Lines
• Line: y = mx + a
`d
• Line: sum of a point and a vector
P = P1 + `d
(where`d is a column vector)
P1
P2
• Line: Affine sum of two points
P = 1P1 + 2P2, where 1 + 2 = 1
P1
Line Segment: For 0 1, 2  1, P lies between P1 and P2
• Line: set of points equidistant from the origin in the
direction of a unit vector.
Plane and Triangle
• Plane: sum of a point and two vectors
P = P1 + `u + b` v
Plane: set of points equidistant
from origin in direction
of a vector.
`v
P1
`u
P3
• Triangle: Affine sum of three points
with i  0
P = 1P1 + 2P2 + 3P3,
where 1 + 2+ 3 = 1
P lies between P1, P2, P3
P1
P2
Generalizing ...
Affine Sum of arbitrary number of points: Convex
Hull
P = 1P1 + 2P2 + … + nPn, where 1 + 2 + … + n = 1 and i 
0
Normal of a Plane
Plane: sum of a point and two vectors
P = P1 + `u + b`v
P - P1 = `u + b`v
`v
P1
`u
If`n is orthogonal to`u and`v (n =`u `v ) :
T
T
T
`n P - P1) = `n `u + b`n `v = 0`n
`v
P1
`u
Implicit Equation of a Plane
T
`n P - P1) = 0
a
x
x1
Let`n = b
P= y
P1 = y1
c
z
z1
Then, the equation of a plane becomes:
a (x - x1) + b (y - y1) + c (z - z1) = 0
`v
P1
`u
`n
`v
ax+by+cz+d=0
P1
`u
Thus, the coefficients of x, y, z in a plane equation define the
normal.
Normal of a Triangle
Normal of the plane containing the triangle (P1, P2 , P3 `n
):
`n = (P2 - P1)  (P3 - P1)
P1
P3
P2
P2
Normal pointing towards you
P1
P3
Normal pointing away from you
P2
P1
P3
• Models constructed with consistent ordering of triangle vertices :
- all clockwise or all counter-clockwise.
• Usually normals point out of the model.
Normal of a Vertex in a Mesh
`nv = (`n1 +`n2 + … +`nk) / k = `ni / k
= average of adjacent triangle normals
`nv
`nk
or better:
`nv =
Ai `ni) / (k Ai) )
= area-weighted average of adjacent triangle normals
`n3
`n1
`n2
Geometry Continued
• Much of last classes’ material in
Appendix A
Matrices
Anm
 a11 a12
a
 21 a22
 a31 a32


 
an1 an 2
 a1m 
 a2 m 
 a3m 

  
 anm 
Sum:
Cnm  Anm  Bnm
cij  aij  bij
A and B must have the same
dimensions
Matrices
Product:
Cn p  Anm Bm p
m
cij   aik bkj
k 1
Identity Matrix:
A and B must have
compatible dimensions
Ann Bnn  Bnn Ann
1 0  0


0 1  0

I 
IA

AI

A
   


0 0  1
Matrices
• Associative T*(U*(V*p)) = (T*U*V)*p
• Distributive T*(u+v) = T*v + T*v
Matrices
Transpose:
Cmn  A nm
cij  a ji
T
If
AT  A
( A  B)  A  B
T
T
( AB)  B A
T
A is symmetric
T
T
T
Matrices
Determinant:
A must be square
 a11 a12  a11 a12
det 

 a11a22 - a21a12

a21 a22  a21 a22
 a11 a12
det a21 a22
a31 a32
a13 
a22

a23   a11
a32

a33 
a23
a33
- a12
a21 a23
a31 a33
 a13
a21 a22
a31 a32
Euclidean transformations
2D Translation
P’
t
P
2D Translation Equation
ty
P
P’
t
y
x
P  ( x, y )
t  (t x , t y )
tx
P'  ( x  t x , y  t y )  Pt
2D Translation using Matrices
ty
P
P’
t
y
x
tx
P  ( x, y )
t  (t x , t y )
t
 x
 x  t x  1 0 t x   
P'  

  y


 y  t y  0 1 t y   1 
 
P
Scaling
P’
P
Scaling Equation
P’
s.y
P
y
P  ( x, y )
P'  ( sx, sy )
P'  s  P
x
s.x
 sx   s 0  x 
P'     
 

 sy  0 s   y 
P'  S  P
S
Rotation
P
P’
Rotation Equations
Counter-clockwise rotation by an angle 
Y’
 x' cos
 y '   sin 
  
P’

P
y
X’
x
P'  R.P
- sin    x 



cos   y 
Degrees of Freedom
 x' cos
 y '   sin 
  
R is 2x2
- sin    x 



cos   y 
4 elements
BUT! There is only 1 degree of freedom: 
The 4 elements must satisfy the following constraints:
R  RT  RT  R  I
det( R )  1
Transformations can be
composed
• Matrix multiplication is associative.
• Combine series of transformations into
one matrix. (example, whiteboard).
• In general, the order matters. (example,
whiteboard).
• 2D Rotations can be interchanged.
Why?
Rotation and Translation
(
cos  -sin  tx
sin 
cos  ty
0
0
x
y
1
)( )
1
Rotation, Scaling and Translation
(
a
-b
tx
b
a
ty
0
0
1
x
y
1
)( )
Rotation about an arbitrary point
• Can translate to origin, rotate, translate
back. (example, whiteboard).
• This is also rotation with one translation.
– Intuitively, amount of rotation is same either
way.
– But a translation is added.
Stretching Equation
P  ( x, y )
P '  ( s x x, s y y )
P’
Sy.y
 sx x  sx
P'     
s
y
0
 y  
P
y
x
Sx.x
0   x
 

sy   y
S
P'  S  P
Linear Transformation
a b   x 
P'  
 

c d   y
 sin 

- cos
 sin 
 sy 
- cos
SVD
cos   s x
0

sin   
 sx
cos  
sy

sin   
 0
0  sin 
s y - cos 

0  sin 

 - cos 
1
cos    x 
 

sin    y 
cos    x 
sin    y 
Affine Transformation
 x
a b tx  
P'  

y

 
c
d
ty

 1 
 
Viewing Position
• Express world in new coordinate
system.
• If origins same, this is done by taking
inner product with new coordinates.
• Otherwise, we must translate.
Simple 3D Rotation
 cos

 - sin 
 0

sin 
cos
0
0  x

0  y


1  z
1
1
1
x
y
z
2
.
.
.
2
2
Rotation about z axis.
Rotates x,y coordinates. Leaves z coordinates fixed.
x
y
z
n
n
n





Full 3D Rotation
 cos

R   - sin 
 0

sin 
cos
0
0  cos b

0  0
1  - sin b
0 sin b  1
0

1
0  0 cos
0 cos b  0 - sin 
0 

sin  
cos 
• Any rotation can be expressed as combination of three
rotations about three axes.
 1 0 0


RR   0 1 0 
 0 0 1


T
• Rows (and columns) of R are
orthonormal vectors.
• R has determinant 1 (not -1).
3D Rotation + Translation
• Just like 2D case
3D Viewing Position
• Rows of rotation matrix correspond to
new coordinate axis.
Rotation about a known axis
• Suppose we want to rotate about u.
• Find R so that u will be the new z axis.
– u is third row of R.
– Second row is anything orthogonal to u.
– Third row is cross-product of first two.
– Make sure matrix has determinant 1.