Definition
Elliptic space is an abstract object and thus an imaginative challenge. The elliptic plane is the easiest instance and is based on spherical geometry. The abstraction involves considering a pair of antipodal points on the sphere to be a single point in the elliptic plane. The breakdown of this identification of two points into one is the premise of H. G. Wells story "The Remarkable Case of Davidson’s Eyes" (1895). Mathematicians commonly refer to the elliptic plane as the real projective plane. Especially in spaces of higher dimension, elliptic geometry is called projective geometry.
As explained by H. S. M. Coxeter
- The name "elliptic" is possibly misleading. It does not imply any direct connection with the curve called an ellipse, but only a rather far-fetched analogy. A central conic is called an ellipse or a hyperbola according as it has no asymptote or two asymptotes. Analogously, a non-Euclidean plane is said to be elliptic or hyperbolic according as each of its lines contains no point at infinity or two points at infinity.
Read more about this topic: Elliptic Geometry
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