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
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
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.