In Euclidean geometry, the lines remain at a constant distance from each other (meaning that a line drawn perpendicular to one line at any point will intersect the other line and the length of the line segment joining the points of intersection remains constant) and are known as parallels.In hyperbolic geometry, by contrast, there are infinitely many lines through A not intersecting l, while in elliptic geometry, any line through A intersects l.Īnother 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 the same plane): 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 l and a point A, which is not on l, there is exactly one line through A that does not intersect l. The essential difference between the metric geometries is the nature of parallel lines. When the metric requirement is relaxed, then there are affine planes associated with the planar algebras, which give rise to kinematic geometries that have also been called non-Euclidean geometry. In the former case, one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises by either replacing the parallel postulate with an alternative, or relaxing the metric requirement. In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry.
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