Bulletin of the Belgian Mathematical Society - Simon Stevin

Hyperfocused Arcs

William E. Cherowitzo and Leanne D. Holder

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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.

Article information

Bull. Belg. Math. Soc. Simon Stevin, Volume 12, Number 5 (2006), 685-696.

First available in Project Euclid: 10 January 2006

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 51E21: Blocking sets, ovals, k-arcs
Secondary: 51A30: Desarguesian and Pappian geometries 05B25: Finite geometries [See also 51D20, 51Exx] 05C70: Factorization, matching, partitioning, covering and packing

hyperovals nets 1-factorizations of complete graphs Desarguesian planes secret sharing schemes


Cherowitzo, William E.; Holder, Leanne D. Hyperfocused Arcs. Bull. Belg. Math. Soc. Simon Stevin 12 (2006), no. 5, 685--696. doi:10.36045/bbms/1136902606. https://projecteuclid.org/euclid.bbms/1136902606

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