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. For example, the sum of the interior angles of any triangle is always greater than 180°. The appearance of this geometry in the nineteenth century stimulated the development of non-Euclidean geometry generally, including hyperbolic geometry.Įlliptic geometry has a variety of properties that differ from those of classical Euclidean plane geometry. Because of this, the elliptic geometry described in this article is sometimes referred to as single elliptic geometry whereas spherical geometry is sometimes referred to as double elliptic geometry. However, unlike in spherical geometry, two lines are usually assumed to intersect at a single point (rather than two). Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold.
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