Absolute Geometry - Relation To Other Geometries

Relation To Other Geometries

The theorems of absolute geometry hold in hyperbolic geometry, which is a non-Euclidean geometry, as well as in Euclidean geometry.

Absolute geometry is inconsistent with elliptic geometry: in that theory, there are no parallel lines at all, but it is a theorem of absolute geometry that parallel lines do exist. More precisely, given any line l and any point P not on l, there is at least one line through P which is parallel to l.

Absolute geometry is an extension of ordered geometry, and thus, all theorems in ordered geometry hold in absolute geometry. The converse is not true. Absolute geometry assumes the first four of Euclid's Axioms (or their equivalents), to be contrasted with affine geometry, which does not assume Euclid's third and fourth axioms. Ordered geometry is a common foundation of both absolute and affine geometry.

It might be imagined that absolute geometry is a rather weak system, but that is not the case. Indeed, in Euclid's Elements, the first 28 Propositions avoid using the parallel postulate, and therefore are valid in absolute geometry. One can also prove in absolute geometry the exterior angle theorem (an exterior angle of a triangle is larger than either of the remote angles), as well as the Saccheri-Legendre theorem, which states that a triangle has at most 180°.