Lecture 2: linear algebra, animation basics

Computer Animation, Rick
Parent, Chapter 2 and
Appendix

Khan Academy online
courses for
 Calculus and Linear Algebra,
geometry, trigonometry
 https://www.khanacademy.org/

b  b
a  ax
ay
x
by

a  b  a
 a   a
sa  sa

b
az
z

b 
a  b  a x  bx
a y  by
a z  bz
 bx
a y  by
az
x
x
x
 ay
sa y
 az
sa z


z

The magnitude (length) of a vector is:
v  v v v
2
x

2
y
Unit vector (magnitude=1.0)
v
v
2
z
a  b  a x bx  a y by  a z bz   ai bi
a  b  a b cos 

How do you find the angle θ between vectors
a and b?
b
θ
a
a  b  a b cos 
 a b 

cos   

a
b




1 a  b

  cos 

a
b


b
θ
a
0 < a·b < 1
a·b = 0
b
-1 < a·b < 0
a·b = 1
θ
a
a·b
a  b  1.0
a  b  cos 
a·b = -1

If a and b are arbitrary (non-unit) vectors,
then the following are still true:
 If θ < 90º then a·b > 0
 If θ = 90º then a·b = 0
 If θ > 90º then a·b < 0

If |u|=1.0 then a·u is the length of the projection of a
onto u
a
u
a·u
i
a  b  ax
bx

j
ay
by
k
az
bz
a  b  a y bz  a z by
a z bx  a x bz
a x by  a y bx

a  b  a b sin 
a  b  area of parallelogram ab
a  b  0 if a and b are parallel
a  b is perpendicular to both a and b,
in the direction defined by the right
hand rule


Dot product produces a scalar
Cross product of two vectors is a vector


Dot product applies to n dimensional vectors
Cross product applies to 3 dimensional vectors

Intuition:
 Dot product shows how much part of the vector a is
in the same direction as vector b
 Cross product is how much part of the vector a is
perpendicular to the vector b

Find the area of the triangle defined by 3D
points a, b, and c
c
b
a
1
area  b  a  c  a 
2
c
c-a
b
a
b-a

An object is at position p with a unit length heading
of h. We want to rotate it so that the heading is
facing some target t. Find a unit axis a and an angle
θ to rotate around.
•
•
p
h
t
h  t  p 
a
h  t  p 
 h  t  p  

  cos 



t

p


t
1
a
•
t-p
θ
•
p
h
Sin
Cos
Tan
Cot
0
0
1
0
Not defined
30
1/2
1/√3
√3
45
1/√2
1/√2
1
1
60
3/2
1/2
√3
1/√3
90
1
0
Not defined
0
3/2
cos2θ + sin2θ = 1
1.0
θ
cos θ
sin θ

Law of Sines:
a
b
c


sin  sin  sin 
 Law of Cosines:
c  a  b  2ab cos 
2
2
2
b
α
γ
c
β
a

5
A=
−1
3
4

5
Det 𝐴 =
−1
3
= 5.4 – (-1.3) = 23
4
1
1
. adj A = .
det(𝐴)
23

Inv 𝐴 =

4
Adj A =
1
−3
5
4 −3
1 5

b  b
a  ax
ay
x
by

b
az
z
a  b  a x bx  a y by  a z bz
L  MN
l11 l12
l
l
21
22

l31 l32
l13   m11 m12


l23   m21 m22
l33  m31 m32
l12  m11n12  m12 n22  m13 n32
m13   n11 n12


m23   n21 n22
m33  n31 n32
n13 

n23 
n33 

Let’s say we have a 3D model that has an array of position
vectors describing its shape
 We store all position vectors: vn where 0≤n≤NumVerts-1

If we want to move the object (translate)
 𝑣𝑛′ = 𝑣𝑛 + 𝑑 (relative offset)
v
'
x
v
'
y
 
v  vx
'
z
 𝑣𝑥′ = 𝑣𝑥 + 𝑑𝑥
 𝑣𝑦′ = 𝑣𝑦 + 𝑑𝑦
 𝑣𝑧′ = 𝑣𝑧 + 𝑑𝑧
vy
 
vz  d x
dy
dz

v
'
x
v 'y
 
v z'  v x
vy
1 0 0
v z 0 1 0  d x
0 0 1

vx  v x 1  v y 0  v z 0  d x
vy  v x 0  v y 1  v z 0  d y
vz  v x 0  v y 0  v z 1  d z

dy
dz


Now, let’s rotate the object in the xy plane by an
angle 𝜃, as if we were spinning it around the z
axis

𝑣𝑥′ = 𝑣𝑥 cos 𝜃 − 𝑣𝑦 sin(𝜃)
𝑣𝑦′ = 𝑣𝑥 𝑠𝑖𝑛 𝜃 + 𝑣𝑦 cos(𝜃)
𝑣𝑧′ = 𝑣𝑧



Note: a positive rotation will rotate the object
counterclockwise when the rotation axis (z) is
pointing towards the observer
𝑣𝑥′ = 𝑣𝑥 cos 𝜃 − 𝑣𝑦 sin(𝜃)
𝑣𝑦′ = 𝑣𝑥 𝑠𝑖𝑛 𝜃 + 𝑣𝑦 cos(𝜃)
𝑣𝑧′ = 𝑣𝑧
y
𝜃 = 30
𝑣𝑥′
𝑣𝑦′
𝑣𝑧′
𝑣𝑥
𝑣𝑦
𝑣𝑧
x
z





𝑣𝑥′ = 𝑣𝑥 cos 𝜃 − 𝑣𝑦 sin(𝜃) + 0𝑣𝑧
𝑣𝑦′ = 𝑣𝑥 𝑠𝑖𝑛 𝜃 + 𝑣𝑦 cos(𝜃) + 0𝑣𝑧
𝑣𝑧′ = 0𝑣𝑥 + 0𝑣𝑦 + 1𝑣𝑧
𝑣𝑥′
𝑣𝑦′
𝑣 ′ = 𝑣. 𝑀
𝑣𝑧′ = 𝑣𝑥
𝑣𝑦
cos(𝜃)
𝑣𝑧 . − sin(𝜃)
0
sin 𝜃
cos(𝜃)
0
0
0
1

1
𝑅𝑥 𝜃 = 0
0
0
cos(𝜃)
−sin(𝜃)

cos(𝜃)
0
𝑅𝑦 𝜃 =
sin(𝜃)
cos(𝜃)
 𝑅𝑧 𝜃 = − sin(𝜃)
0
0
1
0
0
sin(𝜃)
cos(𝜃)
−sin(𝜃)
0
cos(𝜃)
sin 𝜃
cos(𝜃)
0
0
0
1

If we have a vector v, and an x-axis rotation:
v  v . Rx ( )

If we then want to rotate it around y-axis:
v"  v'. R y ( )
v"  v. Rx ( ) R y ( )
v"  v. ( M 1M 2 M 3 M 4 )
v"  v. M total

v  vx
vy
vz

v  v  M
vx  v x a x  v y bx  v z c x
vy  v x a y  v y by  v z c y
vz  v x a z  v y bz  v z c z
a x

M  bx
 cx

ay
by
cy
az 

bz 
c z 
v  v x a  v y b  v z c
vx  v x a x  v y bx  v z c x  d x
vy  v x a y  v y by  v z c y  d y
vz  v x a z  v y bz  v z c z  d z
a, b, c, d are all constants (12 in total)

How do we represent rotation and orientation
together in a matrix?
vx  v x a x  v y bx  v z c x  d x
vy  v x a y  v y by  v z c y  d y
vz  v x a z  v y bz  v z c z  d z
v
'
x
v
'
y
v
'
z
 
1  vx
vy
vz
ax
b
x

1
 cx

d x

ay
by
cy
dy
az
bz
cz
dz
0
0
0

1

3x3 rotation matrix and 3x1 translation vector
combined in a 4x4 matrix (with [0 0 0 1] at the
right)

3D position vector v is changed to [vx vy vz 1]

The line at the right is not used here but it is
necessary when rendering objects as a 2D
image


First, let's look at how projective geometry
works in 2D, before we move on to 3D.
Imagine a projector that is projecting a 2D image
onto a screen.


What happens when the projector goes closer to
the screen?
What is the role of W?

Applying it to 3D
 When W increases the coordinate scales up and when W decreases it scales
down.

Coordinates are said to be correct in 3D, only when W = 1. (convention)






W < 1 everything would look too big
W > 1 everything would look too small
W = 0 division by zero error
W < 0 everything would flip upside down and back-to-front
(15,21,3) => (15/3, 21/3, 3/3) => (5, 7, 1)
1/5 (10, 20, 30, 5) = (2, 4, 6, 1)

Perspective is the phenomenon where an object appears smaller
the further away it is from the camera. (because it is scaled down)

A far-away mountain can appear to be smaller than a cat, if the cat
is close enough to the camera.

Perspective in 3D graphics is implemented by using a transformation
matrix that changes the W element of each vertex.

Z represent the distance, the larger the Z is, the more it needs to be
scaled down.

W effects the scale and it is related with Z

Perspective projection matrix applied to a homogeneous coordinate
1
0

0

0
0
1
0
0
0
0
1
1
0  2  2
0 3 3

0  4  4
   
0  1   4 




After the perspective projection matrix is
applied, each vertex goes under perspective
division.
Converting a homogeneous coordinate back
to W = 1.
¼ (2,3,4,4) = (0.5, 0.75, 1, 1)
After the perspective division, W is discarded.
 Correct 3D coordinate that has been scaled
according to a 3D perspective projection

Technically, homogeneous vectors are 4D
vectors that get projected into the 3D w=1
space
v
x
vy
vz
 vx
vw  
 vw

vy
vw
vz 

vw 


Computer graphics apps commonly use 4x4 homogeneous
matrices
A rigid 4x4 matrix transformation looks like this:
b
ax
b
x

M
 cx

d x

ay
by
cy
dy
az
bz
cz
dz
0

0
0

1
y
c
d
a
x
z
Where a, b, & c are orthogonal unit length vectors
representing orientation, and d is a vector representing
position





The space that an object is defined in is called
object space or local space
The object is located at or near the origin and
is aligned with the xyz axes
The units in this space can be whatever we
choose (i.e. meters, etc)
A 3D object would be stored on disk and in
memory in this coordinate system
When we draw the object, we want to
transform it into another space






We will define a new space called world space or
global space
This space represents a 3D world or scene and may
contain several objects in various locations
Every object in the world needs a matrix that
transforms its vertices from its own object space into
the world space
We call this object’s world matrix
For example, if we have 100 chairs in a room, we only
need to store the object space data for one chair once.
We can use 100 different matrices to transform the
chair model into 100 locations in the world.


The 9 constants make up 3 vectors a, b and c
If we think of the matrix as a transformation
from object to world space
 the a vector is essentially the object’s x-axis
rotated into world space
 b is its y-axis in world space
 and c is its z-axis in world space.

d is the position in world space
1
0
I
0

0




0
1
0
0
0
0
1
0
0
0
0

1
Take one more look at the identity matrix
It’s a axis lines up with x, b lines up with y, and c lines up with
z
Position d is at the origin
Therefore, it represents a transformation with no rotation or
translation
1
0
cos(𝜃)
 𝑅𝑥 𝜃 = 0
0 −sin(𝜃)
cos(𝜃)
0
 𝑅𝑦 𝜃 =
sin(𝜃)
cos(𝜃)
 𝑅𝑧 𝜃 = − sin(𝜃)
0
0
sin(𝜃)
cos(𝜃)
0 −sin(𝜃)
1
0
0 cos(𝜃)
sin 𝜃
cos(𝜃)
0
0
0
1
0
0
1
0 cos( ) sin( )
Rx ( )  
0  sin( ) cos( )

0
0
0
cos( )
 0
R y ( )  
 sin( )

 0
 cos( )
 sin( )
Rz ( )  
 0

 0
0  sin( )
1
0
0 cos( )
0
0
sin( ) 0
cos( ) 0
0
1
0
0
0
0
0

1
0
0
0

1
0
0
0

1

For example, a translation by vector r
followed by a z-axis rotation is:
v x'   cos( ) sin( )
 ' 
v y    sin( ) cos( )
 v z'   0
0
  
0
 1   0
0
0
1
0
0 1
0 0
.
0  0

1  0
0
1
0
0
0 rx  v x 
0 ry  v y 
1 rz   v z 
 
0 1  1 

If the upper 3x3 portion is orthonormal, we
say that 4x4 matrix is rigid
 only translated and rotated (it will not have any
scale or shears which distort the object)



If all row vectors and all column vectors of a
matrix are unit length, that matrix is said to
be orthonormal
This also implies that all vectors are
perpendicular to each other
Orthonormal matrices have some useful
mathematical properties, such as:
 M-1 = MT

If a 4x4 matrix represents a rigid
transformation, then the upper 3x3 portion
will be orthonormal
a  b  c 1
a  bc
b  ca
c  ab

v  vx
vy
v  v  M

vz 1
ax
b
x

M
 cx

d x
vx  v x a x  v y bx  v z c x  d x
vy  v x a y  v y by  v z c y  d y
vz  v x a z  v y bz  v z c z  d z
vw  1
ay
by
cy
dy
az
bz
cz
dz
0
0
0

1
v  v x a  v y b  v z c  d
v  v x a  v y b  v z c  d
y
v=(.5,.5,0,1)
(0,0,0)
x
Local Space
v  v x a  v y b  v z c  d
b
Matrix M
y
y
d
v=(.5,.5,0,1)
(0,0,0)
x
Local Space
(0,0,0)
x
World Space
a
v  v x a  v y b  v z c  d
b
v’
y
y
d
v=(.5,.5,0,1)
(0,0,0)
x
Local Space
(0,0,0)
x
World Space
a

To transform a vector v by matrix M:
v’=v·M

If we want to apply several transformations, we can just
multiply by several matrices:
v’=(((v·M1)·M2)·M3)·M4…

Or we can concatenate the transformations into a single
matrix:
Mtotal=M1·M2·M3·M4…
v’=v·Mtotal

If M transforms v into world space, then M-1
transforms v’ back into local space
v  v  M
v  v  M 1
1
M M  I

We usually transform vertices from some local space where
they are defined into a world space
v’ = v·W

Once in world space, we can perform operations that require
everything to be in the same space (collision detection, high
quality lighting…)

Then, they are transformed into a camera’s space, and then
projected into 2D
v’’ = v’·C-1·P

In simple situations, we can do this all in one step:
v’’ = v·W·C-1·P
while (!finished)
{
UpdateEverything();
DrawEverything();
}


Interactive vs. Non-Interactive
Real Time vs. Non-Real Time






Film
Imax
NTSC TV
PAL TV
HDTV
Computer
24 fps
48 fps
30 fps (interlaced)
25 fps (interlaced)
50-60 fps
>60 fps

Animation tools:





Maya
3D Studio Max
MotionBuilder
Blender
Game/graphics engines:
 OGRE3D
 Unity
 Unreal

Applications of animated characters
 Health
▪ Analyzing walking styles
▪ Analyzing muscle activations
▪ Analyzing joint movement limits
 Ergonomics
▪ Usability of devices
▪ Measuring comfort
 Safety
▪ Simulated worlds for crisis
management
 Entertainment
▪ Games, movies
Different approaches:
 Keyframe animation
 Motion Capture
 Physics-based animation
 Procedural animation

Different levels of character motion

Representation: skeletal model
 A VH is represented by a polyhedral model (or
mesh)
 An underlying skeleton deforms this mesh
▪ Joints, connected by bones
 A pose is defined by the rotations of the joints and
the position of the root joint
 Several standards
▪ H-Anim

ISO standard (ISO/IEC 19774)
for humanoid modelling and
animation

Developed in the late 90’s

Systematic system for
representing humanoids in a
network-enabled 3D graphics
and multimedia environment

http://www.h-anim.org/
Root
Torso
Pelvis
Neck
ShoulderL
ShoulderR
HipL
HipR
Head
ElbowL
ElbowR
KneeL
KneeR
WristL
WristR
AnkleL
AnkleR

Degree of Freedom (DOF)
 A variable φ describing a particular axis or dimension of
movement within a joint

Joints typically have around 1-6 DOFs (φ1…φN)

Changing the DOF values over time results in the
animation of the skeleton

Note: in a mathematical sense, a free rigid body has
6 DOFs: 3 for position and 3 for rotation

Kinematics
 The analysis of motion independent of physical forces.
Kinematics deals with position, velocity, acceleration, and
their rotational counterparts, orientation, angular velocity,
and angular acceleration.

Forward Kinematics
 The process of computing world space geometric data
from DOFs

Inverse Kinematics
 The process of computing a set of DOFs that causes some
world space goal to be met (I.e., place the hand on the
door knob…)

Skeleton
 A pose-able framework of joints arranged in a tree
structure.

Joint




Allows relative movement within the skeleton.
Are essentially 4x4 matrix transformations
Can be rotational, translational, or other
Synonym: bone

Core Joint Data
 DOFs (N floats)
 Local matrix: L
 World matrix: W

Additional Data




Joint offset vector: r
DOF limits (min & max value per DOF)
Type-specific data (rotation/translation axes, constants…)
Tree data (pointers to children, siblings, parent…)
1.
Specify all DOF values for the skeleton (done by higher
level animation system)
2.
Recursively traverse through the hierarchy starting at the
root and use forward kinematics to compute the world
matrices (done by skeleton system)
3.
Use world matrices to deform skin & render (done by skin
system)
Note: the matrices can also be used for other things such as
collision detection, FX, etc.

In the recursive tree traversal, each joint first
computes its local matrix L based on the values of
its DOFs and some formula representative of the
joint type:
Local matrix L = Ljoint(φ1,φ2,…,φN)

Then, world matrix W is computed by
concatenating L with the world matrix of the parent
joint
World matrix W = L · Wparent

It is convenient to have a 3D offset vector r
for every joint which represents its pivot
point relative to its parent’s matrix
L offset
1
0

0

rx
0
1
0
ry
0
0
1
rz
0

0
0

1

It is nice to be able to limit a DOF to some
range (for example, the elbow could be
limited from 0º to 150º)

Usually, in a realistic character, all DOFs will
be limited except the ones controlling the
root

One can then adjust each of the DOFs to specify the pose of
the skeleton

We can define a pose Φ more formally as a vector of N
numbers that maps to a set of DOFs in the skeleton
Φ = [φ1 φ2 … φN]

A pose is a convenient unit that can be manipulated by a
higher level animation system and then handed down to the
skeleton

Usually, each joint will have around 1-6 DOFs, but an entire
character might have 100+ DOFs in the skeleton

Rotational





 Hinge: 1-DOF
 Universal: 2-DOF
 Ball & Socket: 3-DOF
▪ Euler Angles
▪ Quaternions


Free
Screw
Constraint
Etc.
Non-Rigid
 Scale
 Shear
 Etc.
Translational
 Prismatic: 1-DOF
 Translational: 3-DOF (or
any number)
Compound

Design your own...

Rotation around the x-axis:
1
0
L Rx  x   
0

rx
0
cos  x
 sin  x
ry
0
sin  x
cos  x
rz
0
0
0

1

Rotation around the y-axis:
cos  y
 0
L Ry  y   
 sin  y

 rx
0
1
0
ry
 sin  y
0
cos  y
rz
0
0
0

1

Rotation around the z-axis:
 cos  z
 sin 
z

L Rz  z  
 0

 rx
sin  z
cos  z
0
ry
0
0
1
rz
0
0
0

1

Rotation around an arbitrary axis a:
L Ra   
 a x2  c (1  a x2 )

 a x a y (1  c )  a z s
a x a z (1  c )  a y s

rx

a x a y (1  c )  a z s
a y2  c (1  a y2 )
a y a z (1  c )  a x s
ry
a x a z (1  c )  a y s
a y a z (1  c )  a x s
a z2  c (1  a z2 )
rz
0

0
0

1

For a 2-DOF joint that first rotates around x
and then around y:
 cy
s s
x y

L Rxy  x ,  y  
c x s y

 rx

0
cx
 sx
ry
 sy
sxcy
cxc y
rz
0

0
0

1
Different matrices can be formed for
different axis combinations

For a 3-DOF joint that first rotates around x,
y, then z:
c y cz

s s c  c s
x y z
x z
L Rxyz  x ,  y ,  z   
c x s y c z  s x s z

rx


cy sz
sx s y sz  cxcz
cx s y sz  sxcz
ry
Different matrices can be formed for
different axis combinations
 sy
sxcy
cxc y
rz
0
0
0

1

To build a matrix from a set of Euler angles, we just
multiply a sequence of rotation matrices together:
1 0

R x  R y  R z  0 c x
0  s x
0  c y


sx    0
c x   s y

c y cz

  sx s y cz  cx sz
c x s y c z  s x s z

0  s y   cz
 
1 0    s z
0 c y   0
c y sz
sx s y sz  cx cz
cx s y sz  sx cz
 sy 

sxc y 
c x c y 
sz
cz
0
0

0
1

1-DOF translation along an arbitrary axis a:


L Ta t   


rx
1
0
0
 t  ax
0
1
0
ry  t  a y
0
0
1
rz  t  a z
0

0
0

1

For a more general 3-DOF translation:


L Txyz t   


rx
1
0
0
 tx
0
1
0
ry  t y
0
0
1
rz  t z
0
0
0

1

q  qx
qy
qz
qw

q  q x2  q y2  q z2  q w2  1


q  a x sin
2

a y sin
 1  2q y2  2q z2

2q x q y 2q w q z

L Q q  
2q x q z 2q w q y

rx


2
a z sin
2q x q y 2q w q z
1  2q x2  2q z2
2q y q z 2q w q x
ry

2

cos 
2
2q x q z 2q w q y
2q y q z 2q w q x
1  2q x2  2q y2
rz
0

0
0

1