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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?

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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.

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  • $\begingroup$ 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

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