1. Rotation representation and Interpolation
CSE 3541
Matt Boggus

2. Rotation Matrices

3. Interpolating Orientations in 3D
Rotation matrices
Given rotation matrices Mi and time ti, find M(t) such that M(ti)=Mi ux vx nx uy vy ny uz vz nz u v n

4. Flawed Solution Interpolate each entry independently
Example: M0 is identity and M1 is 90o around x-axis
Is the result a rotation matrix?

5. Orientation Representations
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

6. Fixed Angles
Any orientation can be described by composing three rotations, one around each global coordinate axis
Roll, pitch and yaw (perfect for flight simulation)

7. Gimbal Lock
Two or more axes align resulting in a loss of rotation degrees of freedom.

8. Fixed Angles in the Real World
Apollo inertial measurement unit
To “prevent” lock, they added a fourth Gimbal

9. Axis-Angle Q Y A X Z Rotate about given axis Euler’s Rotation Theorem
Fairly easy to interpolate between orientations
Difficult to concatenate rotations

10. Axis-angle to rotation matrixY Z
Concatenate 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 Q A

11. Quaternions (intuitionally)X Y Z Q A
Same as axis-angle, but different form
Still rotate about given axis
Mathematically convenient form
Note: in this form v is a scaled version of the given axis of rotation, A (which is a vector)

12. Quaternion
q = s + xi + yj + zk where i, j, and k are imaginary numbers
Rotation
Quaternions
Identity (no rotation)
1, -1
180 degrees about x-axis
i, -i
180 degrees about y-axis
j, -j
180 degrees about z-axis
k, -k
angle θ, axis (unit vector)

13. Quaternion Arithmetic
Addition
Multiplication
Inner Product
Length

14. Quaternion Arithmetic
Inverse
Identity
Unit quaternion

15. Quaternion Representation and Rotation
Vector
Rotation

16. Unit Quaternion Conversions
Axis-Angle

17. What an animator needs to know
Quaternions
What an animator needs to know
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

18. Rotations within Unity
transform.Rotate(eulerAngles : Vector3,
relativeTo : Space = Space.Self)
Description
Applies a rotation of eulerAngles.z degrees around the z axis, eulerAngles.x degrees around the x axis, and eulerAngles.y degrees around the y axis (in that order).

19. Rotations within Unity
transform.Rotate(axis : Vector3,
angle : float,
relativeTo : Space = Space.Self)
Description
Rotates the transform around axis
by angle degrees.

20. Rotations within Unity
transform.RotateAround(point : Vector3,
axis : Vector3,
angle : float)
Description
Rotates the transform about axis passing through point in world coordinates by angle degrees.
This modifies both the position and the rotation of the transform.

21. Rotations within Unity
Vector3.RotateTowards(current : Vector3, target : Vector3, maxRadiansDelta : float, maxMagnitudeDelta : float) Description Rotates a vector current towards target.

22. Orientation along a curve

23. Orientation on a curve in 2D

24. Orientation on a curve in 2D

25. Orientation on a curve in 2D
q1 and q2 are orthogonal to P’(u)
i.e.
q1 = Rotate(90) * P’(u)
q2 = Rotate(-90) * P’(u)

26. Frenet Frame Global axes: x, y, z Local axes: u, v, w
a moving (coordinate) frame that provides a coordinate system at each point of the curve that is "best adapted" to the curve near that point
Global axes: x, y, z
Local axes: u, v, w
P(s) = parametric curve function with s on [0,1]

27. Frenet Frame
tangent & curvature vector

28. Frenet Frame
tangent & curvature vector

29. local coordinate system
Frenet Frame
local coordinate system
Directly control orientation of object/camera
Use for direction and bank into turn, especially for ground-planar curves (e.g. roads)
w = P’(s)
u = P’’(s) × P’(s)
v = w × u

30. Frenet Frame – sometimes undefined

31. Frenet Frame – discontinuity

32. Other ways to control orientation
Use auxiliary curve to define direction or up vector
Use point P(s+ds) for direction

33. Direction & Up vector
To keep ‘head up’, use y-axis to compute over and up vectors perpendicular to direction vector
v = u × w w
If up vector supplied, use that instead of y-axis
u=w × y-axis
Direction vector

34. Orientation interpolation
Preliminary note:
Remember that
Affects of scale are divided out by the inverse appearing in quaternion rotation
When interpolating quaternions, use UNIT quaternions – otherwise magnitudes can interfere with spacing of results of interpolation

35. Orientation interpolation
Quaternions can be interpolated to produce in-between orientations:
2 problems analogous to issues when interpolating positions:
How to take equidistant steps along orientation path?
How to pass through orientations smoothly (1st order continuous)
And another particular to quaternions: with dual unit quaternion representations, which to use?

36. Dual representation Dual unit quaternion representations: q = –q
For Interpolation between q1 and q2, compute cosine between q1 and q2 and between q1 and –q2; choose smallest angle

37. Interpolating quaternions
Unit quaternions form set of points on 4D sphere
Linearly interpolating unit quaternions: not equally spaced

38. Interpolating quaternions in great arc => equal spacing

39. Interpolating quaternions with equal spacing
where
‘slerp’, sphereical linear interpolation is a function of
the beginning quaternion orientation, q1
the ending quaternion orientation, q2
the interpolant, u
Still a linear order interpolation