Editing Euclidean space (section)

=== Other coordinates ===
[[Image:Repere espace.png|thumb|right|192px|3-dimensional skew coordinates]]
{{main article|Coordinate system}}

As a Euclidean space is an [[affine space]], one can consider an [[affine frame]] on it, which is the same as a Euclidean frame, except that the basis is not required to be orthonormal. This define [[affine coordinates]], sometimes called ''skew coordinates'' for emphasizing that the basis vectors are not pairwise orthogonal.

An [[affine basis]] of a Euclidean space of dimension {{mvar|n}} is a set of {{math|''n'' + 1}} points that are not contained in a hyperplane. An affine basis define [[barycentric coordinates]] for every point.

Many other coordinates systems can be defined on a Euclidean space {{mvar|E}} of dimension {{mvar|n}}, in the following way. Let {{mvar|f}} be a [[homeomorphism]] (or, more often, a [[diffeomorphism]]) from a [[dense subset|dense]] [[open subset]] of {{mvar|E}} to an open subset of <math>\R^n.</math> The ''coordinates'' of a point {{mvar|x}} of {{mvar|E}} are the components of {{math|''f''(''x'')}}. The [[polar coordinate system]] (dimension 2) and the [[spherical coordinate system|spherical]] and [[cylindrical coordinate system|cylindrical]] coordinate systems (dimension 3) are defined this way.

For points that are outside the domain of {{mvar|f}}, coordinates may sometimes be defined as the limit of coordinates of neighbour points, but these coordinates may be not uniquely defined, and may be not continuous in the neighborhood of the point. For example, for the spherical coordinate system, the longitude is not defined at the pole, and on the [[antimeridian]], the longitude passes discontinuously from –180° to +180°.

This way of defining coordinates extends easily to other mathematical structures, and in particular to [[manifold]]s.