On 12/8/06, Fred lunnon <fred.lunnon@gmail.com> wrote:
The following 4-parameter expression generates a planar chart of 4 concyclic vertices for any choice of integer parameters a,b,c,d, excepting just one odd; furthermore it generates precisely _all_ those for which two (or three) pairs of opposite edges are equal, modulo permutations of the vertices A,B,C,D:
[AB, BC, CA, BD, AD, CD] = [a*b, (a*c+b*d)/2, (a*c-b*d)/2, (a*c-b*d)/2, (a*c+b*d)/2, c*d]
I don't know whether a 5-parameter rational parameterisation exists for all concyclic charts.
[I'll have to quote myself, since nobody else seems to be going to --- just in case anybody's still listening out there ...] The above seems to be the best that can be expected in the way of a rational parameterisation: it might just be worth using to search for an ambiguous example. I reason as follows [I can't actually quote a theorem here, and I'd be grateful for any relevant references.] If there exists a rational parameterisation for the general planar chart, i.e. flat tetrahedron in terms of its edge lengths, then the variety in projective 5-space defined by the degree-6 equation in 6 homogeneous variables volume(u,v,w,x,y,z) = 0 has a high-order singularity at some point Q. The only candidates for Q turn out to be a folded umbrella [0,0,0,1,1,1], a folded square [0,1,1,1,1,0], and their symmetries. The parameterisation is then constructed by intersecting the variety with the line joining Q to an arbitrary point P: if the singularity was sufficiently fierce, there will only be one other possible point R of intersection with the surface remaining. Unfortunately, for the umbrella there tirn out to be 2 remaining, and for the square 4. So, no rational solution. Turning to the concyclic case, the construction above has to be modified to intersect a plane joining two singularities Q,Q' and arbitrary P with the 4-variety defined by volume(u,v,w,x,y,z) = 0; u z + v y + w x = 0 (Ptolemy relation). Once again, there turn out to be two resp four points of intersection with the variety remaining. In practice, a quadratically irrational parameterisation might still be utilised to restrict a search for charts --- it's just rather more complicated and less restrictive than a rational one! Fred Lunnon