CPSC 441: Computer Graphics Rotation Representation and Interpolation

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CPSC 441 Computer Graphics Rotation
Representation and Interpolation

• Jinxiang Chai

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Joints and Rotation

• Rotational dofs are widely used in character
  animation

3 translation dofs 48 rotational dofs
1 dof knee
2 dof wrist
3 dof shoulder
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Orientation vs. Rotation 

• Orientation is described relative to some
  reference alignment
• A rotation changes object from one orientation to
  another
• Can represent orientation as a rotation from the
  reference alignment 

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Ideal Orientation Format 

• Represent 3 degrees of freedom with minimum
  number of values
• Allow concatenations of rotations
• Math should be simple and efficient
• concatenation
• interpolation
• rotation

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Outline

• Rotation matrix
• Fixed angle and Euler angle
• Axis angle
• Quaternion

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Matrices as Orientation 

• Matrices just fine, right?
• No
• 9 values to interpolate
• dont interpolate well

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Representation of orientation

• Homogeneous coordinates (review)
• 4X4 matrix used to represent translation,
  scaling, and rotation
• a point in the space is represented as
• Treat all transformations the same so that they
  can be easily combined

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Rotation
old points
New points
rotation matrix
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Interpolation

• In order to move things, we need both
  translation and rotation
• Interpolating the translation is easy, but what
  about rotations?

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Interpolation of Orientation

• How about interpolating each entry of the
  rotation matrix?
• The interpolated matrix might no longer be
  orthonormal, leading to nonsense for the
  inbetween rotations

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Interpolation of Orientation

• Example interpolate linearly from a positive 90
  degree rotation about y axis to a negative 90
  degree rotation about y
• Linearly interpolate each component and halfway
  between, you get this...

Rotate about y-axis with 90
Rotate about y-axis with -90
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Properties of Rotation Matrix

• Easily composed?
• Interpolation?
• Compact representation?

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Properties of Rotation Matrix

• Easily composed? yes
• Interpolation?
• Compact representation?

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Properties of Rotation Matrix

• Easily composed? yes
• Interpolation? not good
• Compact representation?

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Properties of Rotation Matrix

• Easily composed? yes
• Interpolation? not good
• Compact representation?
• - 9 parameters (only needs 3 parameters)

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Outline

• Rotation matrix
• Fixed angle and Euler angle
• Axis angle
• Quaternion

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Fixed angles

• Angles used to rotate about fixed axes
• Orientations are specified by a set of 3 ordered
  parameters that represent 3 ordered rotations
  about fixed axes
• Many possible orderings x-y-z, x-y-x,y-x-z
• - as long as axis does immediately follow
  itself such as x-x-y

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Fixed angles
Ordered triple of rotations about global axes,
any triple can be used that doesnt immediately
repeat an axis, e.g., x-y-z, is fine, so is
x-y-x. But x-x-z is not.
E.g., (qz, qy, qx)
Q Rx(qx). Ry(qy). Rz(qz). P
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Euler Angles vs. Fixed Angles 

• One point of clarification
• Euler angle
• - rotates around local axes
• Fixed angle
• - rotates around world axes
• Rotations are reversed
• - x-y-z Euler angles z-y-x fixed angles

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Euler angle Interpolation

• Interpolating each components separately
• Might have singularity problem
• Halfway between (0, 90, 0) (90, 45, 90)
• Interpolate directly, get (45, 67.5, 45)
• Desired result is (90, 22.5, 90) (verify this!)

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Euler angle concatenation 

• Can't just add or multiply components
• Best way
• Convert to matrices
• Multiply matrices
• Extract Euler angles from resulting matrix
• Not cheap

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Gimbal lock 

• Euler/fixed angles not well-formed
• Different values can give same rotation
• Example with z-y-x fixed angles
• ( 90, 90, 90 ) ( 0, 90, 0 )

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Gimbal lock 

• Euler/fixed angles not well-formed
• Different values can give same rotation
• Example with z-y-x fixed angles
• ( 90, 90, 90 ) ( 0, 90, 0 )

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Gimbal lock 

• Euler/fixed angles not well-formed
• Different values can give same rotation
• Example with z-y-x fixed angles
• ( 90, 90, 90 ) ( 0, 90, 0 )

z
(90,0,0)
y
x
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Gimbal lock 

• Euler/fixed angles not well-formed
• Different values can give same rotation
• Example with z-y-x fixed angles
• ( 90, 90, 90 ) ( 0, 90, 0 )

z
(90,0,0)
(90,90,0)
y
y
x
z
x
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Gimbal lock 

• Euler/fixed angles not well-formed
• Different values can give same rotation
• Example with z-y-x fixed angles
• ( 90, 90, 90 ) ( 0, 90, 0 )

z
(90,0,0)
(90,90,0)
(90,90,90)
z
y
y
x
z
y
x
x
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Gimbal lock

• A Gimbal is a hardware implementation of Euler
  angles used for mounting gyroscopes or expensive
  globes
• Gimbal lock is a basic problem with representing
  3D rotation using Euler angles or fixed angles

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Gimbal lock

• When two rotational axis of an object pointing in
  the same direction, the rotation ends up losing
  one degree of freedom

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Outline

• Rotation matrix
• Fixed angle and Euler angle
• Axis angle
• Quaternion

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Axis angle
Rotate object by q around A
(Ax,Ay,Az,q)
A
q
Y
Z
X
Eulers rotation theorem An arbitrary rotation
may be described by only three parameters.
?
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Axis-angle rotation
Given r Vector in space to rotate n
Unit-length axis in space about which to rotate
q The amount about n to rotate Solve r
The rotated vector
r
r
n
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Axis-angle rotation

• Compute rpar the projection of r along the n
  direction
• rpar (nr)n

r
rpar
r
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Axis-angle rotation

• Compute rperp rperp r-rpar

rperp
r
rpar
r
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Axis-angle rotation

• Compute v a vector perpedicular to rpar and
  rperp v rparxrperp

v
rperp
r
rpar
r
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Axis-angle rotation

• Compute v a vector perpedicular to rpar and
  rperp v rparxrperp

v
rperp
r
rpar
Use rpar, rperp and v, ? to compute the new
vector!
r
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Axis-angle rotation
rperp r (nr) n
q
V n x (r (nr) n) n x r
r
rpar (nr) n
r
n
r rpar rperp
rpar (cos q) rperp (sin q) V (nr) n
cos q(r (nr)n) (sin q) n x r (cos q)r (1
cos q) n (nr) (sin q) n x r
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Axis-angle rotation

• Can interpolate rotation well

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Axis-angle interpolation
1. Interpolate axis from A1 to A2 Rotate axis
about A1 x A2 to get A
A1
q1
A
Y
q
A2
A1 x A2
2. Interpolate angle from q1 to q2 to get q
q2
Z
X
3. Rotate the object by q around A
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Axis-angle rotation

• Can interpolate rotation well
• Compact representation
• Messy to concatenate
• - might need to convert to matrix form

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Outline

• Rotation matrix
• Fixed angle and Euler angle
• Axis angle
• Quaternion

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Quaternion

• Remember complex numbers aib, where i2-1
• Quaternions are a non-commutative extension of
  complex numbers
• Invented by Sir William Hamilton (1843)
• Quaternion
• - Q a bi cj dk where
  i2j2k2ijk-1,ijk,jki,kij
• - Represented as q (w, v) w xi yj
  zk

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Quaternion

• 4 tuple of real numbers w, x, y, z
• Same information as axis angles but in a more
  computational-friendly form

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Quaternion math
Unit quaternion
Multiplication
Non-commutative
Associative
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Quaternion math
Conjugate
Inverse
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Quaternion Example
let
then
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Quaternion Rotation
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Quaternion rotation
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Quaternion Example

• Rotate a point (1,0,0) about y-axis with -90
  degrees

z
(0,0,1)
y
x
(1,0,0)
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Quaternion Example

• Rotate a point (1,0,0) about y-axis with 90
  degrees

z
(0,0,1)
y
x
(1,0,0)
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Quaternion Example

• Rotate a point (1,0,0) about y-axis with 90
  degrees

z
(0,0,1)
y
x
(1,0,0)
Point (1,0,0)
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Quaternion Example

• Rotate a point (1,0,0) about y-axis with 90
  degrees

z
(0,0,1)
y
x
(1,0,0)
Point (1,0,0)
Quaternion rotation
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Quaternion Example

• Rotate a point (1,0,0) about y-axis with 90
  degrees

z
(0,0,1)
y
x
(1,0,0)
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Quaternion composition

• Rotation by p then q is the same as rotation by
  qp

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Matrix Form
For a 3D point (x,y,z)
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Quaternion interpolation
1-angle rotation can be represented by a unit
circle
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Quaternion interpolation
1-angle rotation can be represented by a unit
circle
2-angle rotation can be represented by a unit
sphere
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Quaternion interpolation
1-angle rotation can be represented by a unit
circle
2-angle rotation can be represented by a unit
sphere

• Interpolation means moving on n-D sphere
• Now imagine a 4-D sphere for 3-angle rotation

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Quaternion interpolation

• Moving between two points on the 4D unit
• Sphere
• a unit quaternion at each step - another point
  on the 4D unit sphere
• move with constant angular velocity along the
  great circle between the two points on the 4D
  unit sphere

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Quaternion interpolation
Direct linear interpolation does not
work Linearly interpolated intermediate points
are not uniformly spaced when projected onto the
circle
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Quaternion interpolation
Spherical linear interpolation (SLERP)
Normalize to regain unit quaternion
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Quaternion interpolation
Spherical linear interpolation (SLERP)
Normalize to regain unit quaternion
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Quaternion interpolation
Spherical linear interpolation (SLERP)
Normalize to regain unit quaternion
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Quaternions

• Can interpolate rotation well
• Compact representation
• Also easy to concatenate