elliptic geometry is a non-euclidean geometry, in which, given a line l and a point p outside l, there exists no line parallel to l passing through p, as all lines in elliptic geometry intersect. elliptic geometry has a variety of properties that differ from those of classical euclidean plane geometry. for example, the sum of the angles of any triangle is always greater than 180Â°. ==definitions== in elliptic geometry, two lines perpendicular to a given line must intersect. in fact, the perpendiculars on one side all intersect at the absolute pole of the given line. the perpendiculars on the other side also intersect at a point, which is different from the other absolute pole only in spherical geometry, for in elliptic geometry the poles on either side are the same. there are no antipodal points in elliptic geometry. every point corresponds to an absolute polar line of which it is the absolute pole. any point on this polar forms an absolute conjugate pair with the pole. such a pair of points is orthogonal, and the distance between them is a quadrant.sommerville 1914 p 89 the distance between a pair of points is proportional to the angle between their absolute polars.sommerville 1914 p 101 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.coxeter 1969 94 ==two dimensions== ===the spherical model=== thumb|350px|on a sphere, the sum of the angles of a triangle is not equal to 180Â°. the surface of a sphere is not a euclidean space, but locally the laws of the euclidean geometry are good approximations. in a small triangle on the face of the earth, the sum of the angles is very nearly 180Â°. a simple way to picture elliptic geometry is to look at a globe. neighboring lines of longitude appear to be parallel at the equator, yet they intersect at the poles. more precisely, the surface of a sphere is a model of elliptic geometry if lines are modeled by great circles, and points at each other's antipodes are considered to be the same point. with this identification of antipodal points, the model satisfies euclid's first postulate, which states that two points uniquely determine a line. if the antipodal points were considered to be distinct, as in spherical geometry, then uniqueness would be violated, e.g., the lines of longitude on the earth's surface all pass through both the nort
Classical_geometry Metric_geometry Non-Euclidean_geometry