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Arc (projective geometry)
A (''simple'') arc in finite projective geometry is a set of points which satisfies, in an intuitive way, a feature of ''curved'' figures in continuous geometries. Loosely speaking, they are sets of points that are far from "line-like" in a plane or far from "plane-like" in a three-dimensional space. In this finite setting it is typical to include the number of points in the set in the name, so these simple arcs are called ''k''-arcs. An important generalization of the ''k''-arc concept, also referred to as arcs in the literature, are the (''k'', ''d'')-arcs.
==''k''-arcs in a projective plane==
In a finite projective plane ''π'' (not necessarily Desarguesian) a set ''A'' of ''k'' (''k'' ≥ 3) points such that no three points of ''A'' are collinear (on a line) is called a ''k''-arc. If the plane ''π'' has order ''q'' then ''k'' ≤ ''q'' + 2, however the maximum value of ''k'' can only be achieved if ''q'' is even. In a plane of order ''q'', a (''q'' + 1)-arc is called an oval and, if ''q'' is even, a (''q'' + 2)-arc is called a hyperoval.
A ''k''-arc which can not be extended to a larger arc is called a ''complete arc''. In the Desarguesian projective planes, PG(2,''q''), no ''q''-arc is complete, so they may all be extended to ovals.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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