Spatial Transformations and Math Representations of Rotations

1. Significance

Lack of understanding of rotations and their math representations can lead to serious problems.

Example 1. Gimbal lock and Apollo 11

Example 2. Computer graphics / animation problems

2. Learning Rotations and Math Representations

Examples of Teaching Materials

Text about calculating rotations using trigonometry and matrices

by Rajaa Issa (Essential Mathematics for Computatioal Design)
Very classic text, also explained in lecture by Dr. Chasnov below.

Slides for 2D transformations and matrices in ARCH655, TAMU

Challenges of Learning Rotations

Example 1: Rotation of solar panels in ARCH653 Building Information Modeling

Example 2 - Pilot Studies

Review of rotations and their math representations below.

3. Matrices

Transformations and Matrices

by 3Blue1Brown

This video clip ended with a rotation angle of 90 degree, and the corresponding matrix implicitly contains the information of the angle in the matrix elements. Next, we will see how the angles are explicitly expressed in the matrix in the video below to understand better the relationship between rotations and matrices.

(Note: Affine transformations include Linear tranformations and Translation)

Rotation Matrices

by Dr. Jeffrey Chasnov

After learning matrices for 2D rotations about a single axis, we now look into matrices for 3D rotations about three axes with their corresponding Euler Angles.

4. Euler Angles

Euler Angles and the Euler Rotation Sequence

by Dr. Christopher Lum
(video part 1. Goal of using Euler Angles and sequence through matrices till 3m52s. Continued with detailed implementation.)

"Augmented Reality" at 1m29s !

Animation in Matlab shows Euler Angles rotation sequence starts at 16:10.

Note that Dr. Lum uses the right-hand system, explains the intrinsic rotations (using changed body frame axes) with his aircraft model, does left-multiplication of matrices, and writes the 'transposed' rotation matrices compared to the rotation matrices in https://en.wikipedia.org/wiki/Rotation_matrix -> Basic rotations, which are extrinsic rotations (using fixed world frame axes) with also left-multiplication of matrices. Not sure why he uses the transposed matrices... His animation of the yaw angle = +70 degree at 17m07s seems using the matrix that is not transposed, eventhough the Mathematica coordinate system in the animation is rotated from his North-East-Down coordinate system.

Relation of extrinsic and intrinsic rotations

5. Axis-Angle

Derivation of Rodrigues' Rotation Formula

by Dr. Christopher Lum

Using Rodrigues' Rotation Formula to prove Euler's Rotation Theorem
(A rotation sequence is equivalent to one single rotation)

by Dr. Christopher Lum (video part 3)

6. Quaternions

Visualizing Quaternions (and the Gimbal lock problem showed briefly)

by 3Blue1Brown
The video leads to the actual interactive visualization at
https://eater.net/quaternions

More detailed quaternion calculation

by Dr. Christopher Lum

Try to watch the complete videos above - they are great learning materials for the topics!