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.
Read more about this topic: Arc (projective Geometry)
Famous quotes containing the word plane:
“At the moment when a man openly makes known his difference of opinion from a well-known party leader, the whole world thinks that he must be angry with the latter. Sometimes, however, he is just on the point of ceasing to be angry with him. He ventures to put himself on the same plane as his opponent, and is free from the tortures of suppressed envy.”
—Friedrich Nietzsche (1844–1900)