Rick Parent
Technical Background
Computer Animation

Spaces and Transformations
Left-handed v. right handed
Homogeneous coordinates:
4x4 transformation matrix (TM)
Concatenating TMs
Basic transformations (TMs)
Display pipeline
 x y z 1

Rick Parent Computer Animation

Object Space Data
World space to eye space (TM)
Display
Pipeline
Object space to world space (TM)
World Space Data
Eye Space Data
Perspective (TM)
Clipping
Perspective Divide
Image Space Data
Image space to display space (TM)
Display Space Data
Rick Parent Computer Animation

Representing an orientation
Example: fixed angles - rotate around global axes
Y
Orientation :
    
P '

R z
(

) R y
(

) R x
(

) P
X
Z
Rick Parent Computer Animation

Working with fixed angles and
Rotation Matrices (RMs)
Z
Z
Y
Y
Orthonormalizing a RM
Extracing fixed angles from an orientation
Extracing fixed angles from a RM
X
Making a RM from fixed angles
X
Making a RM from transformed unit coordinate system (TUCS)
Rick Parent Computer Animation

Transformations in pipeline object -> world: often rigid transforms world -> eye: rigid transforms perspective matrix: uses 4 th component of homo. coords perspective divide image -> screen: 2D map to screen coordinates
Clipping: procedure that considers view frustum
Rick Parent Computer Animation

Error considerations
Accumulated round-off error - transform data: transform world data by delta RM update RM by delta RM; apply to object data update angle; form RM; apply to object data orthonormalization rotation matrix: orthogonal, unit-length columns iterate update by taking cross product of 2 vectors scale to unit length considerations of scale miles-to-inches can exeed single precision arithmetic
Rick Parent Computer Animation

Orientation Representation
Rotation matrix
Fixed angles: rotate about global coordinate system
Euler angles: rotate about local coordinate system
Axis-angle: arbitrary axis and angle
Quaternions: mathematically handy axis-angle 4-tuple
Exponential map: 3-tuple version of quaternions
Rick Parent Computer Animation

Transformation Matrix





 i a e m f b n j c g k o l d h p






Rick Parent Computer Animation

Transformation Matrix





 e i a
0 b f
0 j c g k
0 t t t
1 x y z






Rick Parent Computer Animation

Rotation
Matrices

1

 0


0
0 cos sin
0
 
 
 sin cos
0
 
 
0 0
0
0
0
1










 cos
 
0 sin
0
0
0
0
1 sin cos
0
 
0
0

0


0


1 


 cos

 sin
0
0
 
 
 sin cos

 
 
0
0
1
0
0
0
0
1


0

0


Computer Animation Rick Parent

Y
Fixed Angles
X
 
Z
  
P '

R z
(

) R y
(

) R x
(

) P
Fixed order: e.g., x, y, z; also could be x, y, x
Global axes
Rick Parent Computer Animation

Gimbal Lock

0 0
Y
0

Fixed angle: e.g., x, y, z

0 90
Y
0

X
X
Z
Rick Parent
Z
Computer Animation


Gimbal Lock
0 90
Y
0

X
Fixed order of rotations: x, y, z
What do these epsilon rotations do?

0
 
90 0


0 90
 
0


0 90 0
  
Z
Rick Parent Computer Animation


0 90
Y
Gimbal Lock
0
 
90 0
Y
90

X X
Z Z
Interpolating FA representations does not produce intuitive rotation because of gimbal lock
Rick Parent Computer Animation

Euler Angles
Y
    
Prescribed order: e.g., x, y, z or x, y, x
Rotate around (rotated) local axes
Z
 
Note: fixed angles are same as Euler angles in reverse order and vice versa
  
P '

R x
(

) R y
(

) R z
(

) P
X
Rick Parent Computer Animation

Axis-Angle



 
A x
 y z
 
Y
A
X
Z
Rotate about given axis
Euler’s Rotation Theorem
OpenGL
Fairly easy to interpolate between orientations
Difficult to concatenate rotations
Q
Rick Parent Computer Animation

Axis-angle to rotation matrix
Z
Y
A
X
Q Concantenate the following:
Rotate A around z to x-z plane
Rotate A’ around y to x-axis
Rotate theta around x
Undo rotation around y-axis
Undo rotation around z-axis
Rick Parent Computer Animation

Axis-angle to rotation matrix
A
*




 a a a x y z a a a x x x





 a
0 a z y
Rot
(
  x y a x z

)
 a x a y a y a y a z a y
 a z
0 a x a y a z a a az z z




Y
A
 a y a x
0




Z
 cos(

)( I
 ˆ
)
 sin(

) A
*
X
Q
P
P’
Rick Parent Computer Animation

Quaternion
Y
Rot  
A



 cos


2 sin(

2
) * A


Same as axis-angle, but different form
Still rotate about given axis
Mathematically convenient form
 s v
 q
Z
A
X
Note: in this form v is a scaled version of the given axis of rotation, A
Q
Rick Parent Computer Animation

Quaternion Arithmetic
Addition
 s
1
 s
2 v
1
 v
2
  s
1 v
1
  s
2 v
2

Multiplication q
1 q
2

 s
1 s
2
 v
1
 v
2 s
2 v
1
 s
1 v
2
 v
1
 v
2

Inner Product
Length q
1
 q
2
 s
1 s
2
 v
1
 v
2 q
 q
 q
Rick Parent Computer Animation

Quaternion Arithmetic
Inverse q

1 
1 q
2 qq

1  q

1 q


1
 s
 v


0 0 0
 
  
1  q

1 p

1
Unit quaternion q
ˆ  q q
Rick Parent Computer Animation

Quaternion Represention
Vector
Transform

0 v
 v '

Rot q
( v )
 qvq

1
Rick Parent Computer Animation

Quaternion Geometric Operations
Rot q
( v )

Rot
 q
( v )
Rot q
( v )

Rot kq
( v ) v
 
Rot q
( Rot p
( v ))

Rot qp
( v ) v
 
Rot q

1
( Rot q
( v ))
 q

1
( qvq

1
) q
 v
Rick Parent Computer Animation

Unit Quaternion Conversions
Rot  s x y z






1

2
2 xy
2 xz y
2



2
2
2 sz sy z
2
2 xy

2 sz
1

2 x
2 
2 z
2
2 yz

2 sx 1

2
2 xz yz
2 x
2



2
2 sy sx
2 y
2




Axis-Angle
 
2 cos

1
( s )
( x , y , z )
 v / v
Rick Parent Computer Animation

Quaternions
Avoid gimbal lock
Easy to rotate a point
Easy to convert to a rotation matrix
Easy to concatenate – quaternion multiply
Easy to interpolate – interpolate 4-tuples
How about smoothly (in both space and time) interpolate?
Rick Parent Computer Animation