Oval (projective plane)

In projective geometry an oval is a point set in a plane that is defined by incidence properties. The standard examples are the nondegenerate conics. However, a conic is only defined in a pappian plane, whereas an oval may exist in any type of projective plane. In the literature, there are many criteria which imply that an oval is a conic, but there are many examples, both infinite and finite, of ovals in pappian planes which are not conics.

As mentioned, in projective geometry an oval is defined by incidence properties, but in other areas, ovals may be defined to satisfy other criteria, for instance, in differential geometry by differentiability conditions in the real plane.

The higher dimensional analog of an oval is an ovoid in a projective space.

A generalization of the oval concept is an abstract oval, which is a structure that is not necessarily embedded in a projective plane. Indeed, there exist abstract ovals which can not lie in any projective plane.

In a a set $Ω$ of points is called an oval, if:

projective plane

When $| l \cap Ω | = 0$ the line $l$ is an exterior line (or passant),^[1] if $| l \cap Ω | = 1$ a tangent line and if $| l \cap Ω | = 2$ the line is a secant line.

For finite planes (i.e. the set of points is finite) we have a more convenient characterization:^[2]

A set of points in an affine plane satisfying the above definition is called an affine oval.

An affine oval is always a projective oval in the projective closure (adding a line at infinity) of the underlying affine plane.

An oval can also be considered as a special quadratic set.^[3]

and its various degenerations are valid.

Pascal's Theorem

There are many which leave a conic invariant.

projectivities

Abstract ovals[edit]

Following (Bue1966), an abstract oval, also called a B-oval, of order $n$ is a pair $(F,{\mathfrak {G}})$ where $F$ is a set of $n+1$ elements, called points, and ${\mathfrak {G}}$ is a set of involutions acting on $F$ in a sharply quasi 2-transitive way, that is, for any two $(a_{1},a_{2}),(b_{1},b_{2})\in F$ with $a_{i}\neq b_{j}$ for $i,j\in \{1,2\}$ , there exists exactly one $\sigma \in {\mathfrak {G}}$ with $\sigma (a_{1})=a_{2}$ and $\sigma (b_{1})=b_{2}$ . Any oval embedded in a projective plane of order $q$ might be endowed with a structure of an abstract oval of the same order. The converse is, in general, not true for $n\geq 8$ ; indeed, for $n=8$ there are two abstract ovals which may not be embedded in a projective plane, see (Fa1984).

When $n$ is even, a similar construction yields abstract hyperovals, see (Po1997): an abstract hyperoval of order $n$ is a pair $(F,{\mathfrak {G}})$ where $F$ is a set of $n+2$ elements and ${\mathfrak {G}}$ is a set of fixed-point free involutions acting on $F$ such that for any set of four distinct elements $a,b,c,d\in F$ there is exactly one $\sigma \in {\mathfrak {G}}$ with $\sigma (a)=b,\sigma (c)=d$ .

Ovoid (projective geometry)

; Rosenbaum, Ute (1998), Projective Geometry / from foundations to applications, Cambridge University Press, ISBN 978-0-521-48364-3

Beutelspacher, Albrecht

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Oval (projective plane)

projective plane

Pascal's Theorem

projectivities

Abstract ovals[edit]

Ovoid (projective geometry)

Beutelspacher, Albrecht

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O'Keefe, Christine M.

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Finite geometries

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Hirschfeld, J. W. P.

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O'Keefe, Christine M.

O'Keefe, Christine M.

O'Keefe, Christine M.

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Canadian Journal of Mathematics

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Acta Arithmetica

Bill Cherowitzo's Hyperoval Page