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.

  • 1
    $\begingroup$ I would use a Rodrigues formula for the purpose described above with the simple constraint $-\pi/2 < \theta < \pi/2$ $\endgroup$
    – Widawensen
    Jan 24, 2021 at 16:30
  • $\begingroup$ 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. $\endgroup$ Jan 24, 2021 at 22:01


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