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Pascal Ovals in Projective Planes

Published online by Cambridge University Press:  20 November 2018

J. F. Rigby
J. F. Rigby*
Affiliation:
University College, Cardiff, Wales
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A projective plane consists of a set of points and a set of lines, together with a relation of incidence between points and lines, such that

(i) any two distinct points P, Q are incident with a unique line PQ,

(ii) any two distinct lines p, q are incident with a unique point p ∩ q,

(iii) there exist four points, no three of which are incident with the same line.

We shall use the usual geometrical terminology.

An oval in a projective plane π is a set of points of π such that:

(i) no three points of are collinear; this means that a line of π is either a secant of , containing two points of , or a tangent of , containing just one point of , or a non-secant of , containing no point of ;

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 1462 - 1476
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Artzy, R., Pascal's theorem on an oval, Amer. Math. Monthly 75 (1968), 143–146.Google Scholar
2. Buekenhout, F., Plans projectifs à ovoïdes Pascaliens, Arch. Math. 17 (1966), 89–93.Google Scholar
3. Coxeter, H. S. M., The real projective plane (McGraw-Hill, New York, 1949).Google Scholar
4. Qvist, B., Some remarks concerning curves of the second degree in a finite plane, Ann. Acad. Sci. Fenn. Ser. A I Math.-Phys. no. 134 (1952), 27pp.Google Scholar
5. Tits, J., Generalisation des groupes projectifs basées sur leurs propriétés de transitivité, Acad. Roy. Belg. Cl. Sci. Mém. Coll. in 8°27 (1952), no. 2, 115 pp.Google Scholar