In geometry, a Cartesian oval is a plane curve consisting of points that have the same linear combination of distances from two fixed points (foci). These curves are named after French mathematician René Descartes, who used them in optics. Definition Let and be fixed points in the plane, and let and denote the Euclidean distances from these points to a third variable point . Let and be arbitrary real numbers. Then the Cartesian oval is the locus of points S satisfying . The two ovals formed by the four equations and are closely related; together they form a quartic plane curve called the ovals of Descartes. Special cases In the equation , when and the resulting shape is an ellipse. In the limiting case in which P and Q coincide, the ellipse becomes a circle. When it is a limaçon of Pascal. If and the equation gives a branch of a hyperbola and thus is not a closed oval. Polynomial equation The set of points satisfying the quartic polynomial equation where is the distance between the two fixed foci and , forms two ovals, the sets of points satisfying the two of the following four equations that have real solutions. The two ovals are generally disjoint, except in the case that or belongs to them. At least one of the two perpendiculars to through points and cuts this quartic curve in four real points; it follows from this that they are necessarily nested, with at least one of the two points and contained in the interiors of both of them. For a different parametrization and resulting quartic, see Lawrence. Applications in optics As Descartes discovered, Cartesian ovals may be used in lens design. By choosing the ratio of distances from and to match the ratio of sines in Snell's law, and using the surface of revolution of one of these ovals, it is possible to design a so-called aplanatic lens, that has no spherical aberration. Additionally, if a spherical wavefront is refracted through a spherical lens, or reflected from a concave spherical surface, the refracted or reflected wavefront takes on the shape of a Cartesian oval. The caustic formed by spherical aberration in this case may therefore be described as the evolute of a Cartesian oval. History The ovals of Descartes were first studied by René Descartes in 1637, in connection with their applications in optics. These curves were also