From: rusin@vesuvius.math.niu.edu (Dave Rusin) Newsgroups: sci.math,sci.math.num-analysis,sci.math.symbolic Subject: Re: Parametrization of a curve Date: 31 Mar 1998 03:01:17 GMT In article <352009B6.5B6A02C0@alcor.cs.purdue.edu>, Cassiano Durand wrote: >Anybody has any suggestion of how to parametrize the curve >obtained by intersecting a sphere and a cylinder like >defined by the system > > x^2 + y^2 + z^2 = a^2 > > y^2 + (z-b)^2 = c^2 Yes and no. The curve projects one-to-one onto its image in the x-y plane; thus it's necessary and sufficient to be able to parameterize the latter. But that curve is of the form y^2=quartic in x. If you don't mind extracting square roots, you can parameterize half the curve at a time using x as your parameter. On the other hand, if you wanted a parameterization using rational or trig functions, you're out of luck because this is an elliptic curve (i.e. of genus 1). Nearest thing to a replacement is the Weierstrass Pe-function. (Here I'm excluding a few degenerate cases e.g. b=0 in which a rational parameterizaion is possible.) dave ============================================================================== From: rusin@math.niu.edu (Dave Rusin) Newsgroups: sci.math.research Subject: Re: Yet another elliptic curve Date: 6 May 1998 03:59:59 GMT In article <6imvau$pv0$1@rzsun02.rrz.uni-hamburg.de>, Hauke Reddmann wrote: >Can y**2=(x**2-2*x-3)*(3*x**2+2*x+1) be completely >parametrized, or does it only contain sporadic >rational points? If you mean "parameterized by rational functions", then you're asking if the curve is of genus zero, and the answer is no. The transformation 3 X + 22 Y {x = --------, y = 136 ---------} X - 38 2 (X - 38) for example renders this curve in minimal form 2 3 Y = X + 20 X - 144 I don't know what "sporadic rational points" means. The group of rational points has torsion subgroup of order 2 (generated by (X,Y)=(4,0) ) and torsion-free part of rank 1 (generated by (X,Y)=(72, 612) ) so that in particular there are infinitely many rational points which may all be computed recursively. dave