Editing Erlangen program (section)

==Homogeneous spaces==

In other words, the "traditional spaces" are [[homogeneous space]]s; but not for a uniquely determined group. Changing the group changes the appropriate geometric language.

In today's language, the groups concerned in classical geometry are all very well known as [[Lie group]]s: the [[classical groups]]. The specific relationships are quite simply described, using technical language.

===Examples===

For example, the group of [[projective geometry]] in ''n'' real-valued dimensions is the symmetry group of ''n''-dimensional real [[projective space]] (the [[general linear group]] of degree {{nowrap|''n'' + 1}}, quotiented by [[Scalar matrix|scalar matrices]]). The [[affine group]] will be the subgroup respecting (mapping to itself, not fixing pointwise) the chosen [[hyperplane at infinity]]. This subgroup has a known structure ([[semidirect product]] of the [[general linear group]] of degree ''n'' with the subgroup of [[translation (geometry)|translation]]s). This description then tells us which properties are 'affine'. In Euclidean plane geometry terms, being a parallelogram is affine since affine transformations always take one parallelogram to another one. Being a circle is not affine since an affine shear will take a circle into an ellipse.

To explain accurately the relationship between affine and Euclidean geometry, we now need to pin down the group of Euclidean geometry within the affine group. The [[Euclidean group]] is in fact (using the previous description of the affine group) the semi-direct product of the orthogonal (rotation and reflection) group with the translations.  (See [[Klein geometry]] for more details.)