# 2-sphere in Euclidean space

From Diffgeom

## Definition

This is the special case of the sphere in Euclidean space where the sphere has dimension 2 and the Euclidean space it is embedded in has dimension 3.

### Implicit and parametric descriptions

Degree of generality | Implicit description | What the parameters mean | Parametric description | What the additional parameters mean | Comment |
---|---|---|---|---|---|

Arbitrary | are the coordinates of the center and is the radius of the sphere | and play roles analogous to the azimuthal and polar angles. See spherical polar coordinates. | |||

Up to translations, i.e., given any 2-sphere in Euclidean space, we can do a translation and bring it into this form | is the radius. | . | and play roles analogous to the azimuthal and polar angles. See spherical polar coordinates. | We have used a translation to move the center of the sphere to the origin. | |

Up to all rigid motions (translations, rotations, reflections) | is the radius. We have used a translation to move the center of the sphere to the origin. | . | and play roles analogous to the azimuthal and polar angles. See spherical polar coordinates. | Since the sphere has rotational and reflection symmetry, allowing freedom of rotation does not result in any simplification of the equation. | |

Up to all similarity transformations (transformations, rotations, reflections, scaling) | no parameters any more. This is the unit 2-sphere centered at the origin. | . | and play roles analogous to the azimuthal and polar angles. See spherical polar coordinates. | We've scaled the sphere to unit radius. |