Rotations and quaternions
Jiří Chmelík, Marek Trtík
PA199

►Rotation matrix
►
►Euler angles
►Tait-Bryan angles
►
►Axis-angle representation
►
►Quaternions
►Rotations via quaternions
►Quaternion derivative
Outline

Rotation matrix
3


Euler angles
4
“Line of nodes”

Euler angles
5
“Line of nodes”

Euler angles
6
“Line of nodes”

Euler angles
7
“Line of nodes”

Euler angles
8
“Line of nodes”

Euler angles
9


Euler angles
10


Euler angles
11


Euler angles
12


Euler angles
13


Euler angles
14
More intuition is in video: [5]

Euler angles: gimbal lock
15


►We can use 3 angles to express any orientation of an object in 3D space:
     public class Orientation {
         float alpha;
         float beta;
         float gamma;
};
►Pros:
►Low memory footprint.
►Easy to understand.
►Cons:
►Suffers from the gimbal lock.
►Slow conversion to matrix representation (sin and cos for each angle).
Euler angles representation
16

Tait-Bryan angles
17


Axis-angle rotation
18


Axis-angle rotation
19


Axis-angle rotation
20


Axis-Angle to rotation Matrix
21


Linear interpolation (lerp)
22


Spherical linear interpolation (slerp)
23


►We can use axis-angle to express any orientation of object in 3D space.
     public class Orientation {
         float angle;
         Vector3 unitAxis;
};
►Pros:
►Fast conversion to matrix representation (sine and cosine for one angle).
►We can use lerp and slerp.
►Easy to understand.
►Low memory footprint.
►Cons:
►Complicated composition of rotations (often solved via other rep.).
Axis-angle representation
24

Quaternions
25


Quaternions
26


Rotation via quaternion
27
TADYà

Rotation via quaternion
28


Quaternions and other representations
29


Linear interpolation (lerp)
30


Spherical linear interpolation (slerp)
31


Quaternion derivative
32


Quaternion derivative
33


Quaternion derivative
34


►We can use quaternions to express any orientation of object in 3D space.
     public class Orientation {   // Equals to a quaternion q=(s,u), |q|=1.
         float s;  // the scalar part
         Vector3 u; // the vector part
};
►Pros:
►Low memory footprint.
►Fast conversion to rotation matrix (no need to compute cosine & sine).
►Fast composition of rotations (just multiply the quaternions).
►We can use lerp and slerp.
►Cons:
►Less human readable.
Quaternion representation
35

►There is no single winner – each representation has pros and cons.
►
►Examples:
►When specifying a rotation along a coordinate axis (e.g., world Z), then Euler angles are a good
choice.
►When specifying a rotation along non-coordinate axis, then axis-angle representation is a good
choice.
►When composing rotations of some joint of a skeleton, then quaternions can do it quickly.
►When rotations must be composed with other transformations, then use matrix representation.
►
►Game engine should provide all representations and conversions between them.
What representation to use?
36

►[1] D.M.Mount; Lecture notes CMSC 425 Game Programming; University of Maryland, 2016.
►[2] E.B.Dam,M.Koch,M.Lillholm; Quaternions, Interpolation and Animation; TechnicalReport
DIKU-TR-98/5, University of Copenhagen, 1998.
►[3] M.C.Nechyba; Lecture notes EEL6667: Kinematics, Dynamics and Control of Robot Manipulators -
Introduction to quaternions; 2003,
https://mil.ufl.edu/nechyba/www/__eel6667.f2003/course_materials.html.
►[4] Y.-B. Jia; Lecture notes (Com S 477/577 Notes) Quaternions; 2018.
►[5] Euler angles: https://www.youtube.com/watch?v=A6lf8t9WXn8
References
37