CQ3

Sep. 24

As we've learned, representing rotations using quaternions and SLERPing between them is the superior method for blending 2 rotational keyframes together. Here are two naive alternatives:

1. LERPing Rotation matrices — We represent the keyframes as rotation matrices $R_1,R_2$ and compute $uR_2+(1−u)R_1$ for a desired $u\in[0,1]$ and apply the resulting matrix to any object(s) to be rotated.
2. LERPing Euler angles — We represent the keyframes as Euler angles $(α_1,β_1,γ_1)$ and $(α_2,β_2,γ_2)$ and compute a new set of Euler angles as $(uα2+(1−u)α1,uβ2+(1−u)β1,uγ2+(1−u)γ1)$ for a desired $u\in[0,1]$ and use this new set of angles to do the rotation (via converting this new set of angles to a rotation matrix).

Q1

In 1-2 sentences each, explain what problems arise with each of the two naive approaches given.

(1) LERPing Rotation Matrices: This method distorts rotations due to loss of orthogonality and can create non-smooth transitions.

(2) LERPing Euler Angles: This approach suffers from gimbal lock and can result in sudden jumps and nonlinear paths.

Q2

In 1-2 more sentences, explain why interpolatory motion achieved via Quaternion SLERPing is more desirable to the motion achieved via Euler angle LERPing.

Quaternion SLERPing is more desirable because it provides a smooth, continuous interpolation along the shortest path on the unit sphere, avoiding issues like gimbal lock. This results in valid and natural-looking rotations, enhancing the overall quality of animations and simulations.