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Elliptic geometry, a special case of Riemannian 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 interior angles of any triangle is always greater than 180°. ==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 line forms an ''absolute conjugate pair'' with the pole. Such a pair of points is ''orthogonal'', and the distance between them is a ''quadrant''.〔Duncan Sommerville (1914) ''The Elements of Non-Euclidean Geometry'', chapter 3 Elliptic geometry, pp 88 to 122, George Bell & Sons〕 The distance between a pair of points is proportional to the angle between their absolute polars.〔 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〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Elliptic geometry」の詳細全文を読む スポンサード リンク
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