Arc (projective Geometry) - (k, d)-arcs in A Projective Plane

(k, d)-arcs in A Projective Plane

A (k, d)-arc (k, d > 1) in a finite projective plane π (not necessarily Desarguesian) is a set, A of k points of such that each line intersects A in at most d points, and there is at least one line that does intersect A in d points. A (k, 2)-arc is a k-arc and may be referred to as simply an arc if the size is not a concern.

The number of points k of a (k, d)-arc A in a projective plane of order q is at most qd + d − q. When equality occurs, one calls A a maximal arc.

Hyperovals are maximal arcs. Complete arcs need not be maximal arcs.