# Describe a set of rotations with a constraint on largest change

I'm looking for a sensible way to parametrize the full set of possible 3D rotations that satisfy the property that they don't rotate any vector in $$\mathbb{R}^3$$ by more than some given angle $$\alpha$$, where $$\alpha < \pi/2$$.

If we were to use rotation matrices to describe the rotation, then the set can be perhaps(?) defined as:

$$S = \{R \in SO(3) \mid \max\{\cos^{-1}(e_1^TRe_1), \cos^{-1}(e_2^TRe_2), \cos^{-1}(e_3^TRe_3)\} < \alpha\},$$ where $${e_i}$$ are the canonical basis vectors - though I'm not at all sure if these three vectors provide a sufficient constraint.

I'm struggling to rephrase the initial constraint on $$\alpha$$ into constraints on the rotation parameters such as Euler angles, or quaternions for non-trivial rotations.

The goal is to be able to sample and apply these rotations randomly, but also to understand what the above property means analytically.

• I would use a Rodrigues formula for the purpose described above with the simple constraint $-\pi/2 < \theta < \pi/2$ Jan 24, 2021 at 16:30
• OK. But just to clarify, $\alpha$ is the constraint on the largest change. The comment about $\pi/2$ is only there to clarify that $\alpha$ will never exceed 90 degrees, but it will be typically much smaller. Jan 24, 2021 at 22:01