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 by , 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 by , one gets an oval which is not a conic. The implicit curve 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 with elements let For and and coprime, the set 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 is an oval in a pappian projective plane and the group of projectivities which leave invariant is 3-transitive, i.e. for 2 triples of points there exists a projectivity with . In the finite case 2-transitive is sufficient. An oval in a pappian projective plane of characteristic is a conic if and only if for any point of a tangent there is an involutory perspectivity (symmetry) with center which leaves invariant. If is an oval in a finite Desarguesian (pappian) projective plane of odd order, , then is a conic (Segre's theorem, ). This implies that, after a possible change of coordinates, every oval of with odd has the parametrization : 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 is a ()-arc, in other words, a set of points, no three collinear. Ovals in the Desarguesian (pappian) projective plane for odd are just the nonsingular conics. However,