A non-Euclidean geometry is characterized by a non-vanishing Riemann curvature tensorâ€”it is the study of shapes and constructions that do not map directly to any n-dimensional Euclidean system. Examples of non-Euclidean geometries include the hyperbolic and elliptic geometry, which are contrasted with a Euclidean geometry. The essential difference between Euclidean and non-Euclidean geometry is the nature of parallel lines. Euclid's fifth postulate, the parallel postulate, is equivalent to Playfair's postulate, which states that, within a two-dimensional plane, for any given line â„“ and a point A, which is not on â„“, there is exactly one line through A that does not intersect â„“. In hyperbolic geometry, by contrast, there are infinitely many lines through A not intersecting â„“, while in elliptic geometry, any line through A intersects â„“ (see the entries on hyperbolic geometry, elliptic geometry, and absolute geometry for more information).
Another way to describe the differences between these geometries is to consider two straight lines indefinitely extended in a two-dimensional plane that are both perpendicular to a third line:
- In Euclidean geometry the lines remain at a constant distance from each other, and are known as parallels.
- In hyperbolic geometry they "curve away" from each other, increasing in distance as one moves further from the points of intersection with the common perpendicular; these lines are often called ultraparallels.
- In elliptic geometry the lines "curve toward" each other and eventually intersect.