Hyperfocused Arcs

Abstract

An arc in a projective plane whose secants meet some exterior line in the minimum number of points is said to be hyperfocused on that line. This is similar to a concept introduced by Simmons in the design of a geometric secret sharing scheme. Drake and Keating have recently rediscovered the idea under the guise of hyperovals in nets. We will answer two open questions raised by Drake and Keating. We also provide a classification of small hyperfocused arcs in Desarguesian planes using the graph-theoretic concept of 1-factorizations of complete graphs.