Oval (projective plane)
In projective geometry an oval is a circle-like pointset (curve) in a plane that is defined by incidence propertes. 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.
When | l ∩ Ω | = 0 the line l is an exterior line (or passant), if | l ∩ Ω | = 1 a tangent line and if | l ∩ 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:
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.
In any pappian projective plane there exist nondegenerate projective conic sections and any nondegenerate projective conic section is an oval. This statement can be verified by a straightforward calculation for any of the conics (such as the parabola or hyperbola).
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