[Computer Animation] 3D Rotation and Orientation

3D Rotation and Orientation

Orientation vs. Rotation

• Rotation
  • Circular movement
• Orientation
  • The state of being oriented.
  • Given a coordinate system, the orientation of an object can be represented as a rotation from a reference pose.
• Analogy
  • (point : vector) is similar to (orientation : rotation)
  • Both represent a sort of (state : movement)

 

What is 3D Rotation?

• Many different ways to describe
  • Rotation matrices
  • Euler angles
  • Axis-angle
  • Rotation vector
  • Unit quaternions

 

Euler's Rotation Theorem

• The general displacement of a rigid body with one point fixed is a rotation about some axis. - Leonhard Euler (1707-1783)
• In other words, 
  • Arbitary 3D rotation euqals to one rotation around an axis.
  • Any 3D rotation leaves one vector unchanged

 

3D rotation

• Given two arbitrary orientations of a rigid object,

 

 

 

Axis-Angles (or Rotation Vector)

Axis-Angles

 

Rotation Vector

 

Rotation vector → Axis-Angle?

• Rotation Vector
  • $q = (3, 6, 7)$
• Axis-Angle
  • axis : $\hat{v} = ?$
    • $(\frac{3}{\sqrt{94}}, \frac{6}{\sqrt{94}}, \frac{7}{\sqrt{94}})$
  • angle : $\theta = ?$
    • $\sqrt{94}$

 

Rotate a point $p$ by an axis-angle?

 

Axis-Angle → Rotation Matrix

 

Step 1: Rotate $u$ on the z-axis

• Rotate $u$ onto the z-axis

 

• Rotate $u$ onto the z-axis
  • $u'$ : Project $u$ onto the yz-plane to compute angle $\alpha$
  • $u''$ : Rotate $u$ about the x-axis by angle $\alpha$
  • Rotate $u''$ onto the z-axis

 

• Rotate $u'$ about the x-axis onto the yz-plan
  • Let $u = (a, b, c)$ and thus $u' = (0, b, c)$
  • Let $u_{z} = (0, 0, 1)$

 

• Rotate $u'$ about the x-axis onto the yz-plan
  • Since we know both $cos(\alpha)$ and $sin(\alpha)$, the rotation matrix can be obtained.

 

Axis-Angle → Rotation Matrix

1. Rotate the axis, $u$ on the z-axis
   • $R_{y}(\beta) · R_{x}(\alpha)$
2. Rotate $p$ around the z-axis
   • $R_{z}(\theta)$
3. Rotate to the original axis
   • $R_{x}^{-1}(\alpha) · R_{y}^{-1}(\beta)$

 

Rotate a point $p$ by an axis-angle?

 

Orientations by Axis-Angles

 

 

Euler Angles

Euler Angles

• A sequence of three elemental rotations
  • elemental rotations: rotation about x, y, or z-axis
  • e.g) $(\alpha, \beta, γ) → R_{z}(γ)R_{y}(\beta)R_{x}(\alpha)$
• Rotation about three orthogonal axes
  • 12 combinations
    • XYZ, XYX, XZY, XZX
    • YZX, YZY, YXZ, YXY
    • ZXY, ZXZ, ZYX, ZYZ

 

Gimbal

• Hardware implementation of Euler Angles
• Aircraft, Camera

 

Gimbal Lock

• Coincidence of inner most and outmost gimbals' rotation axes
• Loss of degree of freedom
• https://compsci290-s2016.github.io/CoursePage/Materials/EulerAnglesViz/index.html

 

Pros and Cons

• Cons
  • Many-to-one mappong.
  • Discontinuity.
  • Gimbal lock problem.
• Pros
  • Simple and intuitive.
  • The most dominant method for end-user interface.

 

 

Rotation Matrices (SO(3) matrices)

SO(3) Group

• Rotation matrices from SO(3)

 

• One matrix in SO(3) corresponds to one unique 3D orientation.
  • One-to-one mapping between SO(3) group and 3D orientations.

 

3D Rotation about z-axis, x-axis, and y-axis

 

Rotation about an arbitary-axis?

 

Axis-angle → Rotation Matrix

 

Rotation matrix $q_{r}$ that rotates from $q_{1}$ to $q_{2}$

 

 

Orientations and Rotations

• Orientations by SO(3) matrices
  • SO(3) matrices are good for representing 3D orientations because they provide one-to-one mapping and no singularity
• Rotations by SO(3) matrices
  • Matrix representation allow to handle the rotation and translation in a uniform way.
    • e.g) 4x4 homogeneous matrices.
  • But, SO(3) matrices cannot represent more than 360 degree rotational movement.

 

Disadvantages of using 3x3 matrices

• Many parameters
  • 9 real numbers for one matrix.
• Many calculations
  • Matrix addition, multiplication, and inverse.
• Normalization problem
  • Repeated calculations yield numerical errors.
  • At some point, the result matrix may not be in SO(3).
  • You need to normalize the matrix regularly during a series of calculations.
• Interpolation?
  • Slerp is difficult.

 

 

Quaternions

Complex numbers and Quaternions

• William Rowan Hamilton (1805-1865)

 

Review the complex numbers in 2D

 

Can $S^{2}$ include all the possible orientations in 3D space?

 

What about $S^{3}$?

 

Unit Quaternions

 

Rotations and Unit Quaternions

 

Question

• What is the unit quaternion for 90˚ rotation along x-axis?

$ = \begin{bmatrix} cos45˚ \\ 1·sin45˚ \\ 0·sin45˚ \\ 0·sin45˚ \end{bmatrix} $ $ = \begin{bmatrix} \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \\ 0 \\ 0 \end{bmatrix} $

 

Antipodal Equivalence

• $q = (w, x, y, z)$ and $-q = (-w, -x, -y, -z) $ represent the same rotation
• 2-to-1 mapping between $S^{3}$ and $SO(3)$

 

Unit Quaternion Algebra

• Identity
  • $q = (1, 0, 0, 0)$
• Multiplication
  • $q_{1}q_{2} = (w_{1}, v_{1})(w_{2}, v_{2}) = (w_{1}w_{2} - v_{1}v_{2}, w_{1}v_{2} + w_{2}v_{1} + v_{1}×v_{2})$
• Inverse
  • $q^{-1} = \frac{(w, -x, -y, -z)}{(w^{2} + x^{2} + y^{2} + z^{2})} = \frac{(-w, x, y, z)}{(w^{2} + x^{2} + y^{2}+ z^{2})}$
• Unit quaternion space is
  • closed under multiplication and inverse,
  • but not closed under addition and subtraction

 

Rotation Quaternion

• Rotation quaternion, $q_{r}$ that rotates from $q_{1}$ to $q_{2}$

 

• Rotate a point $p$ by quaternion $q$

 

Rotate a point $p$ by an axis-angle

 

 

 

Quaternion Exp and Log

Exponential Map

• Map from a trangent plane to the manifold.

 

Quaternion Exp and Log

 

Rotation Quaternion

 

Exp and Log

 

Interpolation

 

Rotation Matrix vs. Unit Quaternion

• Equivalent in many aspects
  • No singularity (continuous)
  • Exp & Log
  • Special tangent space
• Why quaternions?
  • Fewer parameters
  • Simpler algebra
  • Easy to to fix numerical error
• Why rotation matrices?
  • One-to-one correspondence
  • Handle rotation and translation in a uniform way
    • Eg) 4x4 homogeneous matrices

 

Rotation Conversions

• In theory, conversion between any representations is possible
• In practice, conversion is not simple because of different conventions.
• Quaternion to Matrix