![]() ![]() The elliptic plane is the real projective plane provided with a metric: Kepler and Desargues used the gnomonic projection to relate a plane σ to points on a hemisphere tangent to it. 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. A central conic is called an ellipse or a hyperbola according as it has no asymptote or two asymptotes. It does not imply any direct connection with the curve called an ellipse, but only a rather far-fetched analogy. The name "elliptic" is possibly misleading. The distance between a pair of points is proportional to the angle between their absolute polars. Such a pair of points is orthogonal, and the distance between them is a quadrant. Any point on this polar line forms an absolute conjugate pair with the pole. The reason for doing this is that it allows elliptic geometry to satisfy the axiom that there is a unique line passing through any two points.Įvery point corresponds to an absolute polar line of which it is the absolute pole. ![]() For example, this is achieved in the hyperspherical model (described below) by making the "points" in our geometry actually be pairs of opposite points on a sphere. This is because there are no antipodal points in elliptic geometry. However, unlike in spherical geometry, the poles on either side are the same. The perpendiculars on the other side also intersect at a point. In fact, the perpendiculars on one side all intersect at a single point called the absolute pole of that line. In elliptic geometry, two lines perpendicular to a given line must intersect. ![]()
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