For example, we can represent a rotation in the xy plane as

$$R=\left\{ R(\theta)=\begin{pmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{pmatrix}, 0 \leq \theta \leq 2\pi \right\}$$

But this matrix R can be embedded in a rotation in 3D, for instance, to represent the rotation around the z axis as

$$R=\left\{ R(\theta)=\begin{pmatrix} \cos(\theta) & -\sin(\theta) & 0 \\ \sin(\theta) & \cos(\theta) & 0 \\ \ 0 & 0 & 1 \end{pmatrix}, 0 \leq \theta \leq 2\pi \right\}$$

What happens to the underlying geometry when you embed a smaller matrix into a larger matrix. Specifically, we know that a rotation in 2D is isomorphic to the circle, what happens to this geometry?


It just means that the z-axis remains fixed, and the coordinates (x,y) rotate with relation to it. Similarly, there are variation of 3D rotations, with fixed x-axis, or fixed y-axis.

  • $\begingroup$ Sorry I was looking something that dealt with fibration and sorts, maybe you know something about that? $\endgroup$ – Carlos - the Mongoose - Danger Jan 22 '15 at 20:38

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.