Animation and Games Development
242-515, Semester 1, 2014-2015
7. Basic Maths
• Objective
o to give some background on the course
242-515 AGD: 7. Basic Maths
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Overview
1.
2.
3.
4.
5.
6.
7.
Vectors
Matricies
The Dot Product
The Cross Product
Transformations as Matricies
Multiple Coordinate Spaces
Models in a Scene Graph
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1. Vectors
• A vector is a magnitude (size) and a direction.
o drawn as an arrow
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• Two vectors V and W are added by placing the
beginning of W at the end of V
• Subtraction reverses the second vector
W
V
V+W
W
V–W
V
–W
V
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Triangle Rule for Subtraction
• Place c and d tail-to-tail.
• d – c is the vector from the head of c to the head
of d
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6
7
• Vectors are added and subtracted
component-wise:
V  W  V1  W1 , V2  W2 ,
, Vn  Wn
V  W  V1  W1 ,V2  W2 ,
,Vn  Wn
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Vector Magnitude
• The magnitude of a vector is a scalar (a number):
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Normalization
• A normalized vector always has unit length.
• To normalize a non-zero vector, divide by its
magnitude.
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Normalize [12, -5]:
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Normalization as a Diagram
unit circle
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2. Matrices
• A matrix is an array of numbers arranged in rows
and columns
o A matrix with n rows and m columns is an n  m matrix
o M is a 2  3 matrix
o If n = m, the matrix is a square matrix
1 2 3
M

 4 5 6 
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• The transpose of a matrix M is denoted MT and has
its rows and columns exchanged:
1 2 3
M

 4 5 6 
242-515 AGD: 7. Basic Maths
1 4


T
M  2 5


 3 6 
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• An n-dimensional vector V can be thought of as an
n  1 column matrix:
V1 
 
V2 
V  V1 , V2 , , Vn   
 
 
Vn 
• Or a 1  n row matrix:
V T  V1 V2
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Vn 
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Matrix Multiplication
• The entries of the product of two matricies A and B
are calculated by:
m
 AB ij   Aik Bkj
k 1
• The number of columns in A must equal the number
of rows in B.
• If A is a n  m matrix, and B is an m  p matrix, then
AB is an n  p matrix.
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• Example:
 2 3   2 1   8 13
M



1 1  4 5  6 6 
M 11  2   2   3  4

M 12  2 1  3   5 
 13
8
M 21  1  2    1  4  6
M 22  11   1   5  
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• In three dimensions, the product of a matrix and a
column vector looks like:
 M 11

 M 21

 M 31
M 12
M 22
M 32
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M 13  Vx   M 11Vx  M 12Vy  M 13Vz 
  

M 23  Vy    M 21Vx  M 22Vy  M 23Vz 
  

M 33  Vz   M 31Vx  M 32V y  M 33Vz 
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Identity Matrix In
For any n  n matrix M, the product with the
identity matrix is M itself
o I nM = M
o MIn = M
Invertible
• An n  n matrix M is invertible if there exists another
matrix G such that
0
1 0


0
0 1
MG  GM  I n  





0 0
1 
• The inverse of M is denoted M-1
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Determinant
• The determinant of a square matrix M is denoted
det M or |M|
• A matrix is invertible if its determinant is not zero.
• For a 2  2 matrix,
a b  a b
det 
 ad  bc

 c d  c d
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• The determinant of a 3  3 matrix is
m11
m12
m13
m21
m22
m23  m11
m31
m32
m33
m22
m23
m32
m33
 m12
m21
m23
m31
m33
 m13
m21
m22
m31
m32
 m11  m22 m33  m23 m32   m12  m21m33  m23m31 
 m13  m21m32  m22 m31 
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Inverse
• Explicit formulas exist for matrix inverses
o These are good for small matrices, but other methods are
generally used for larger matrices
o In computer graphics, we usually deal with 2  2,
3  3, and a special form of 4  4 matrices
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3. The Dot Product
• The dot product is a product between two vectors
that produces a magnitude (a number).
• The dot product between two
n-dimensional vectors V and W is given by
n
V  W   VW
i i
i 1
• In three dimensions,
V  W  VxWx  VyWy  VzWz
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Dot product a· b is the magnitude of the projection of
the b vector onto the a vector.
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Sign of the Dot Product
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• The dot product equation:
V  W  V W cos a
oa is the angle between the two vectors
o That's why the dot product is 0 between perpendicular
vectors
o If V and W are unit vectors, the dot product is 1 for parallel
vectors pointing in the same direction, -1 for opposite
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Dot Product for Finding Angles
• The dot product can be used to find the angle
between two vertors a and b.
• First normalize a and b.
^
• The angle between them is cos-1 â . b
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A Line as a Vector
• A line in 3D space can be represented by
R(t) = S + t V
o S is a point on the line, and V is the direction along which
the line runs. R is a point on the line, t is a magnitude.
V
t
R
S
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Distance to a Line
• Distance d from a point P to the line R(t)= S + t V
P
P S
d
S
P  S  V
V
V
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• Use Pythagorean theorem:
 P  S  V 
d   P  S  

V


2
2
2
• Taking square root,
d   P  S 2 
 P  S   V  2
V2
• If V is unit length, then V 2 = 1
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4. The Cross Product
• The cross product between two vectors a and b
produces a vector (a x b) which is perpendicular to
both.
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• The cross product between V and W is
V  W  VyWz  VzWy , VzWx  VxWz , VxWy VyWx
• One way of remembering this equation is as the
determinant:
ˆi
ˆj
kˆ
V  W  Vx
Vy
Vz
Wx Wy Wz
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• It can also be expressed as a matrix-vector product
 0

V  W   Vz

 Vy
Vz
0
Vx
Vy  Wx 
 
Vx  Wy 
 
0  Wz 
• The perpendicularity property means that:
 V  W  V  0
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 V  W  W  0
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• The cross product equation is:
V  W  V W sin a
• This is equal to the area of the parallelogram
formed by V and W:
V
||V|| sin a
a
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W
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• Here's why:
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b
a
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What about the axb Direction?
• Does the vector a x b point up or down from the
plane of a and b?
• Place the tail of b at the head of a.
• Using your right hand, curl your fingers in the
direction of the vectors.
• Your thumb points in the
direction of a x b
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• Reversing the order of the vectors negates the cross
product:
W  V  V  W
o the cross product is anti-commutative
• If a x b = 0, then a is parallel to b.
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5. Transformations as Matricies
• A complex transformation can be built by
combining simple transformations, of the types:
o translations
o scaling
o rotations
• Transformations are combined by multiplying their
matrices together.
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2D Translation
y
y
(x+dx, y+dy)
(x, y)
dy
(0,0)
x
dx
x
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2D Scaling
• Resizes an object in each dimension about origin
s

x
x
0






x 0










0
s

y
y
0
  y
 
y
y
(x, y)
x
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(sxx, syy)
x
43
2D Rotation
• Rotate counter-clockwise about the origin by angle 
 
 

x
cos

sin
x
0

















y
sin
cos
y
0





y
y
(x, y)
x
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(x', y')

x
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Homogenous Transformation Matricies
• An 4x4 homogenous matrix can store 3D
transformations.
• This makes matrix inversion and combinations of
transformations simpler.
• It also makes it possible to define perspective
transformations.
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3D Translation
Translation by (Δx, Δy, Δz):
x
1 0 0 
0 1 0 

y

T
x,
y,
z
0 0 1 
z


0 0 0 1
T1(
x,
y,
z)T(
x,
y,
z)
3D Scaling
Scaling about the origin
sx 0 0
0 s 0
y

Ssx , sy , sz  
 0 0 sz

0 0 0
0

0
0

1
1 1 1
S (sx , sy , sz )  S , , 
 sx sy sz 


1
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3D Rotations
• z-axis rotation
M z -rotate
 cos 

 sin 

 0

 0
 sin 
0
0
cos 
0
0
0
1
0
0
0
1







• Rotates about the z-axis through the angle 
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• Rotations about the x-, y- axes
M x -rotate
M y -rotate
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






1
0
0
0
0
cos 
 sin 
0
0
sin 
cos 
0
0
0
0
1
0
sin 
0
1
0
0
0
cos 
0
0
0
1
 cos 

 0

  sin 

 0














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Combining Transformations
• The combined 4  4 homogenenous matrix is very
common in computer graphics:
 R11

 R21
M
 R31

 0

R12
R13
R22
R23
R32
R33
0
0
Tx 

Ty 

Tz 

1

o R is a 3  3 rotation matrix, and T is a translation vector
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• The inverse of this 4  4 matrix is 'simple':



M 1  




R 1
0
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  R111
 
1
R T   R211

  R311
 
1   0
 
R121
R131
1
R22
1
R23
R321
R331
0
0
  R 1T  x 

1
  R T y 

  R 1T  z 


1

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Transformations are not Commutative
y
y
Translation followed
by a rotation
x
x
y
Rotation followed
by a translation
(the meaning of the
combined matrix)
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x
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Concatenation of Transformations
• How do we decompose a rotation about an
arbitrary point into a sequence of basic
transformations?
y
y
x
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x
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Rotating about an Arbitrary Point
y
translate
to origin
y
(x0, y0)
x
x
rotate
y
y
(xo, yo)
x
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translate
back to (xo, yo)
a TRT sequence
x
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• The orderings of the three operations is important
o the rotation must be in between the two translations, or the
end result will be different
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T then R ≠ R then T
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6. Multiple Coordinate Spaces
• A point will be in a different location in different
coordinate spaces:
y
v
P
(2,3)
y
u
P
(1,2)
u
x
P in the xy coordinate space
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v
x
P in the uv coordinate space
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• We use different coordinate spaces because some
graphics operations are easier in one coordinate
space instead of another.
o e.g. rotating the hand of a robot model (see later)
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Some Coordinate Spaces
• World space
o the global coordinate system
o use it to keep track of every object in the game
• Object space
o
o
o
o
The origin is located at object’s origin
The space's axes are aligned with the object's orientation
The object's coordinates are defined using this space
Called a local space in jME
• Camera space
o Camera’s “object” space
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Robot in World Space
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Robot’s Object Space
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Object and World Space
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Camera Space
Object space for the
viewer, represented by a
camera.
The camera space is
used to project the 3D
world onto the computer
screen.
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Where the camera is
What the camera shows
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Camera Space Terminology
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Frustum Clipping
Clipping: objects partially on screen
Occlusion: objects hidden behind another object
7. Models in a Scene Graph
• In a scene graph, a complex model is usually made
up of simpler parts, connected together in a
parent-child relationship.
Scene Graph
The 3D Model
top half arm
bottom half arm
hand
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Multiple Object Spaces
• Each model part will have its own object space.
• All these object spaces make the transformation of
model parts (e.g. hand rotation) much easier.
y
object space
x for top half arm
y
y
x
object space
for bottom half arm
x object space
for hand
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• Each object space can be thought of as a 'joint',
allowing the model part in that space to be
rotated.
y
x object space
for top half arm
y
x
y
x
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object space
= elbow joint
for bottom half arm
object space
= hand joint
for hand
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Example: hand rotation
• Rotate the hand's object space (the hand joint).
y
x
= hand joint
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• This rotation would be harder in the global space,
since the rotation would need to be preceded by a
translation of the wrist to the origin, rotation, then
translation back
o a TRT sequence
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Child Rotation
• Another advantage of the scene graph is that a
rotation of an model part (e.g. the lower-half arm)
affects it and all of its children (e.g. the hand).
y
= elbow joint
x
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