1. UMBC Graphics for Games
Quaternion Rotation
UMBC Graphics for Games

2. Representing Rotation

3. Rotation Matrices Rotation can be expressed as an orthogonal matrix
Every orthogonal matrix is a rotation and/or mirroring
New X axis
New Y axis
New Z axis

4. Rotation Matrices Rotation can be expressed as an orthogonal matrix
Every orthogonal matrix is a rotation and/or mirroring
Fits with other transformation matrices
Translation, Scaling, Reflection, Perspective
Interpolations between rotation matrices are not rotations
Interpolates endpoint positions

5. Euler Angles Rotate around series of axes Subject to gimbal lock
Roll, Pitch, Yaw
Intuitive, pitch/yaw form common for first person controllers
Yaw = turn left/right
Pitch = look up/down
Roll = rotate around view
Subject to gimbal lock
First axis lines up with last axis, lose ability to rotate freely
Interpolates along lines of longitude/latitude
Can be weird near poles

6. Quaternions Extension of complex numbers
Interpolations on great circle
Related Geometric Algebra also works
Though not as popular in games

7. Complex Numbers

8. Imaginary number i Define imaginary number i where
Complex number has real and imaginary components
Interpret as points on the 2D plane
Conjugate: negate the imaginary part

9. Complex Arithmetic Addition: add components Multiplication
Component algebra:
Multiply magnitudes, add angles:

10. Complex Transformations
Point
Translation = addition
Scale = multiply by real scalar
Rotate = multiply by unit-length complex (magnitude = 1)
Component form is identical to matrix rotation

11. Basic Rotations
Pure imaginary = rotation by 90

12. Quaternions

13. Quaternion basics
First created by Hamilton in 1843 for physics rotations
Define orthogonal imaginary axes
Multiplication order matters, positive in ijk order

14. Quaternion Math
Like complex has real + imaginary, quaternion is real + vector
Conjugate negates vector part
Addition by component

15. Quaternion Multiplication
By component
Vector form

16. Quaternion Transformations
Point is pure quaternion
Translation = addition of pure quaternion
Scale = multiply by real scalar
Rotation = sandwiched multiply by unit quaternion and conjugate
Where
For one point, algebraically optimized version with cross products
For more points, cheaper to convert to matrix

17. Basic Rotations
Rotation
Rotate by 0º
Rotate by 180º around vector

18. Interpolation
Interpolate along great circle between two vectors / quaternions
Normalized Linear Interpolation (NLERP)
Cheap, but speeds up in the middle
Spherical Linear Interpolation (SLERP)
Interpolate in equal-angular steps