Cheshire Cat

The union of the two quadric surfaces, the sphere and the ellipse are the "cat" and the intersection is the "grin". In Lewis Carroll's Alice and Wonderland the Cheshire cat vanished leaving only the grin. Actually the intersection curve of these two quadric surfaces is known as a Quadric Surface Intersection Curve, QSIC, and, as some of the simplest space curves, are part of classical mathematics. But QSIC such as the "grin" here are essentially elliptic curves and there are so many other possibilities that only recently have we had the computing power to adequately deal with them.

In 2008 a solution to the problem was proposed by L.Dupont, D.Lazard, S.Lazard and S.Petitjean, with a 65 page paper which goes into detail on all possible QSIC. But this has been implemented nicely in blackbox software which is freely available on their server.

I have proposed a much simpler method, although not easily reduced to a blackbox program. While the DLLP solution uses mostly exact computations, I simplify the problem greatly by using floating point (decimal) computations. In the Cheshire cat example the quadrics are

f = x2 + y2 + z2 - 16 = 0
g = 57 - 12 x + 4 x2 + y2 + 64 z + 16 z2 = 0

I use a numerical version of a classical exact technique to show that this QSIC is equivalent to the plane cubic curve

which in turn is equivalent to the numerical elliptic cubic

y2 = 52.8574 - 62.9491 x + 29.0454 x2 - 4.4002 x3

A Mathematica package to make these reductions in plane curves is available.

The upper and lower halves of this symmetric curve are easily parameterized by taking the square root. Then since the equivalences above are given by rational functions which have been calculated we can derive a parameterization of the "grin" by composition. We get complicated rational functions of a single variable with one square root in the numerator and denominator.

A complete discussion of this method is given in the paper Real QSIC.