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.
Definition of an oval In a projective plane a set Ω of points is called an oval, if: Any line l meets Ω in at most two points, and For any point P ∈ Ω there exists exactly one tangent line t through P, i.e., t ∩ Ω = {P}. 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:
For a finite projective plane of order n (i.e. any line contains n + 1 points) a set Ω of points is an oval if and only if |Ω| = n + 1 and no three points are collinear (on a common line). 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.
Examples
Conic sections
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). Non-degenerate conics are ovals with special properties:
Pascal's Theorem and its various degenerations are valid. There are many projectivities which leave a conic invariant.
Ovals which are not conics in the real plane If one glues one half of a circle and a half of an ellipse smoothly together, one gets a non-conic oval. If one takes the inhomogeneous representation of a conic oval as a parabola plus a point at infinity and replaces the expression x2 by x4, one gets an oval which is not a conic. If one takes the inhomogeneous representation of a conic oval as a hyperbola plus two points at infinity and replaces the expression 1/x by 1/x3, one gets an oval which is not a conic. The implicit curve x4 + y4 = 1 is a non conic oval. in a finite plane of even order In a finite pappian plane of even order a nondegenerate conic has a nucleus (a single point through which every tangent passes), which can be exchanged with any point of the conic to obtain an oval which is not a conic. For the field K = GF(2m) with 2m elements let
Ω = { ( x , y ) ∈ K 2 | y = x 2 k } ∪ { ( ∞ ) } {\displaystyle \Omega =\{(x,y)\in K^{2}\;|y=x^{2^{k}}\;\}\;\cup \;\{(\infty )\}}
For k ∈ {2,...,m − 1} and k and m coprime, the set Ω is an oval which is not a conic. Further finite examples can be found here:
Criteria for an oval to be a conic For an oval to be a conic the oval and/or the plane has to fulfill additional conditions. Here are some results:
An oval in an arbitrary projective plane, which fulfills the incidence condition of Pascal's theorem or the 5-point degeneration of it, is a nondegenerate conic. If Ω is an oval in a pappian projective plane and the group of projectivities which leave Ω invariant is 3-transitive, i.e. for 2 triples A1, A2, A3 ; B1, B2, B3 of points there exists a projectivity π with π(Ai) = Bi, i = 1,2,3. In the finite case 2-transitive is sufficient. An oval Ω in a pappian projective plane of characteristic ≠ 2 is a conic if and only if for any point P of a tangent there is an involutory perspectivity (symmetry) with center P which leaves Ω invariant. If Ω is an oval in a finite Desarguesian (pappian) projective plane of odd order, PG(2, q), then Ω is a conic by Segre's theorem.). This implies that, after a possible change of coordinates, every oval of PG(2, q) with q odd has the parametrization :
{ ( t , t 2 , 1 ) ∣ t ∈ G F ( q ) } ∪ { ( 0 , 1 , 0 ) } . {\displaystyle \{(t,t^{2},1)\mid t\in GF(q)\}\cup \{(0,1,0)\}.}
For topological ovals the following simple criteria holds:
5. Any closed oval of the complex projective plane is a conic.
Further results on ovals in finite planes An oval in a finite projective plane of order q is a (q + 1, 2)-arc, in other words, a set of q + 1 points, no three collinear. Ovals in the Desarguesian (pappian) projective plane PG(2, q) for q odd are just the nonsingular conics. However, ovals in PG(2, q) for q even have not yet been classified. In an arbitrary finite projective plane of odd order q, no sets with more points than q + 1, no three of which are collinear, exist, as first pointed out by Bose in a 1947 paper on applications of this sort of mathematics to the statistical design of experiments. Furthermore, by Qvist's theorem, through any point not on an oval there pass either zero or two tangent lines of that oval.
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