Ingeometry, aCartesianoval is aplane curve consisting of points that have the samelinear combination of distances from two fixed points (foci). These curves are named after French mathematicianRené Descartes, who used them inoptics.
LetP andQ be fixed points in the plane, and letd(P,S) andd(Q,S) denote theEuclidean distances from these points to a third variable pointS. Letm anda be arbitraryreal numbers. Then the Cartesian oval is thelocus of pointsS satisfyingd(P,S) +m d(Q,S) =a. The two ovals formed by the four equationsd(P,S) +m d(Q,S) = ± a andd(P,S) −m d(Q,S) = ± a are closely related; together they form aquartic plane curve called theovals of Descartes.[1]
In the equationd(P,S) +m d(Q,S) =a, whenm = 1 anda > d(P,Q) the resulting shape is anellipse. In thelimiting case in whichP andQ coincide, the ellipse becomes acircle. When it is alimaçon of Pascal. If and the equation gives a branch of ahyperbola and thus is not a closed oval.
Theset of points(x,y) satisfying thequartic polynomial equation[1][2]
wherec is the distance between the two fixedfociP = (0, 0) andQ = (c, 0), forms two ovals, the sets of points satisfying two of the following four equations
that have real solutions. The two ovals are generallydisjoint, except in the case thatP orQ belongs to them. At least one of the two perpendiculars toPQ through pointsP andQ cuts this quartic curve in four real points; it follows from this that they are necessarily nested, with at least one of the two pointsP andQ contained in the interiors of both of them.[2] For a different parametrization and resulting quartic, see Lawrence.[3]
As Descartes discovered, Cartesian ovals may be used inlens design. By choosing the ratio of distances fromP andQ to match the ratio ofsines inSnell's law, and using thesurface of revolution of one of these ovals, it is possible to design a so-calledaplanatic lens, that has nospherical aberration.[4]
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. Thecaustic formed by spherical aberration in this case may therefore be described as theevolute of a Cartesian oval.[5]
The ovals of Descartes were first studied by René Descartes in 1637, in connection with their applications in optics.
These curves were also studied byNewton beginning in 1664. One method of drawing certain specific Cartesian ovals, already used by Descartes, is analogous to a standard construction of anellipse by a pinned thread. If one stretches a thread from a pin at onefocus to wrap around a pin at a second focus, and ties the free end of the thread to a pen, the path taken by the pen, when the thread is stretched tight, forms a Cartesian oval with a 2:1 ratio between the distances from the two foci.[6] However, Newton rejected such constructions as insufficientlyrigorous.[7] He defined the oval as the solution to adifferential equation, constructed itssubnormals, and again investigated its optical properties.[8]
The French mathematicianMichel Chasles discovered in the 19th century that, if a Cartesian oval is defined by two pointsP andQ, then there is in general a third pointR on the same line such that the same oval is also defined by any pair of these three points.[2]
James Clerk Maxwell rediscovered these curves, generalized them to curves defined by keeping constant the weighted sum of distances from three or more foci, and wrote a paper titledObservations on Circumscribed Figures Having a Plurality of Foci, and Radii of Various Proportions. An account of his results, titledOn the description of oval curves, and those having a plurality of foci, was written byJ.D. Forbes and presented to theRoyal Society of Edinburgh in 1846, when Maxwell was at the young age of 14 (almost 15).[6][9][10]
