Arc (projective geometry)

In mathematics, a (k, d)-arc (k, d > 1) in a finite projective plane π (not necessarily Desarguesian) is a set of k points of $\pi$ 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. When d = 2 it is typical to refer to a (k, d)-arc as simply a k-arc or an arc if the size is not a concern.

Special cases

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.

(q + 1, 2)-arcs are precisely the ovals and (q + 2, 2)-arcs are precisely the hyperovals (which can only occur for even q). Notice that hyperovals are maximal arcs.

A k-arc which can not be extended to a larger arc is called a complete arc. Complete arcs need not be maximal arcs.

References

• Hirschfeld, J.W.P. (1979), Projective Geometries over Finite Fields, New York: Oxford University Press, ISBN 0-19-853526-0 

External links

• C.M. O'Keefe (2001), "Arc", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1556080104, http://www.encyclopediaofmath.org/index.php?title=Arc&oldid=15518