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In mathematics, an oval in a projective plane is a set of points, no three collinear, such that there is a unique tangent line at each point (a ''tangent line'' is defined as a line meeting the point set at only one point, also known as a ''1-secant''). If the projective plane is finite of order ''q'', then the tangent condition can be replaced by the condition that the set contains ''q''+1 points. In other words, an oval in a finite projective plane of order ''q'' is a (''q''+1,2)-arc, or a set of ''q''+1 points, no three collinear. Ovals in the Desarguesian projective plane PG(2,''q'') for ''q'' odd are just the nonsingular conics. Ovals in PG(2,''q'') for ''q'' even have not yet been classified. Ovals may exist in non-Desarguesian planes, and even more abstract ovals are defined which cannot be embedded in any projective plane. ==Odd ''q''== In a finite projective plane of odd order ''q'', no sets with more points than ''q'' + 1, no three of which are collinear, exist, as first pointed out by Bose in a 1947 paper on applications of this sort of mathematics to statistical design of experiments. Due to Segre's theorem , every oval in PG(2, ''q'') with ''q'' odd, is projectively equivalent to a nonsingular conic in the plane. This implies that, after a possible change of coordinates, every oval of PG(2, ''q'') with ''q'' odd has the parametrization : : 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「Oval (projective plane)」の詳細全文を読む スポンサード リンク
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