Cartesian oval

These curves are named after French mathematician René Descartes, who used them in optics.

The set of points (x, y) satisfying the quartic polynomial equation[1][2] where c is the distance

between the two fixed foci P = (0, 0) and Q = (c, 0), forms two ovals, the sets of points satisfying the two of the following four equations that have real solutions.

The caustic formed by spherical aberration in this case may therefore be described as the evolute of a Cartesian oval.

One method of drawing certain specific Cartesian ovals, already used by Descartes, is analogous to a standard construction of an ellipse by a pinned thread.

[7] He defined the oval as the solution to a differential equation, constructed its subnormals, and again investigated its optical properties.

[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 titled Observations on Circumscribed Figures Having a Plurality of Foci, and Radii of Various Proportions.

An account of his results, titled On the description of oval curves, and those having a plurality of foci, was written by J.D.

Forbes and presented to the Royal Society of Edinburgh in 1846, when Maxwell was at the young age of 14 (almost 15).