# Changes

## Torus in 3-space

, 08:16, 31 July 2007
no edit summary
* $y = r_1 \sin \alpha + r_2 \sin \alpha \cos \beta$
* $z = r_2 \sin \beta$

Both $\alpha$ and $\beta$ are modulo $2\pi$.
==Abstract structure==

Topologically, and even differentially, the torus is isomorphic to the direct product of [[circle]]s, viz $S^1 \times S^1$. One coordinate describes the position of the center on the base circle, and the other coordinate describes the position of the point on that particular small circle. The Cartesian parametric equations given above make this explicit.

However, the metric structure on the torus is very different from that on $S^1 \times S^1$ with the direct product metric. The latter is actually the [[flat torus]], which cannot be embedded in $\R^3$, but needs to be embedded in $\R^4$.

==Structure and symmetry==

The isometry group of the torus is the semidirect product of the following two groups:

* The group $S^1$, which acts on the first coordinate by rotating the base circle.
* The group of order two which reflects the whole configuration about the $xy$-plane