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The Steiner conic or more precisely Steiner's generation of a conic, named after the Swiss mathematician Jakob Steiner, is an alternative method to define a non-degenerate projective conic section in a projective plane over a field. The usual definition of a conic uses a quadratic form (see Quadric (projective geometry)). Another alternative definition of a conic uses a ''hyperbolic polarity''. It is due to ''K. G. C. von Staudt'' and sometimes called a von Staudt conic. The disadvantage of von Staudt's definition is that it only works when the underlying field has odd characteristic (i.e., ). == Definition of a Steiner conic == *Given two pencils of lines at two points (all lines containing and resp.) and a projective but not perspective mapping of onto . Then the intersection points of corresponding lines form a non-degenerate projective conic section〔, p. 80〕 〔''Jacob Steiner’s Vorlesungen über synthetische Geometrie'', B. G. Teubner, Leipzig 1867 (from Google Books: ((German) Part II follows Part I )) Part II, pg. 96〕 (figure 1) A ''perspective'' mapping of a pencil onto a pencil is a bijection (1-1 correspondence) such that corresponding lines intersect on a fixed line , which is called the ''axis'' of the perspectivity (figure 2). A ''projective'' mapping is a finite sequence of perspective mappings. Examples of commonly used fields are the real numbers , the rational numbers or the complex numbers . The construction also works over finite fields, providing examples in finite projective planes. ''Remark:'' The fundamental theorem for projective planes states, that a projective mapping in a projective plane over a field (pappian plane) is uniquely determined by prescribing the images of three lines. That means that, for the Steiner generation of a conic section, besides two points only the images of 3 lines have to be given. These 5 items (2 points, 3 lines) uniquely determine the conic section. ''Remark:'' The notation "perspective" is due to the dual statement: The projection of the points on a line from a center onto a line is called a perspectivity (see below).〔 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「Steiner conic」の詳細全文を読む スポンサード リンク
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