Subject: Rational function parametric representations
From: Barton Willis
Date: Tue, 22 Sep 2009 11:44:35 -0500
To find a rational function parametric representation of a folium of
Descartes,
there is (the cute trick) of using the substitution y = x * p:
(%i25) [x^3 + y^3 - 3*a*x*y, y = x * p];
(%o25) [y^3-3*a*x*y+x^3,y=p*x]
(%i26) algsys(%,[x,y]);
(%o26) [[x=(3*a*p)/(p^3+1),y=(3*a*p^2)/(p^3+1)],[x=0,y=0]]
What is the generalization of this trick? Sometime ago, I saw an
article on this, but I can't find it. Maybe I gave up to soon, but
a web search on "parametric representation" and "parametric
representation algorithm" didn't locate the paper.
Maple has a function that tries to find these parametric representations,
but the user documentation doesn't give a reference to the algorithm.
Barton