From: Dave Rusin <rusin@math.niu.edu>
Subject: Re:  bezier curve intersection
Date: Fri, 2 Mar 2001 10:20:41 -0600 (CST)
To: bertl@sbox.tu-graz.ac.at
Summary: Computing intersection of two Bezier curves

Are you asking about something more sophisticated than simply solving
the simultaneous pair of cubic equations  x1(t)=x2(u), y1(t)=y2(u)  ?
One could, for example, search globally for solutions by using
elimination (Grobner bases, say) to reduce to a polynomial in one variable;
or if what is desired is a numerical determination of an intersection,
one has recourse to Newton's method.

[deletia --djr]

(The generic case solves for  t  in terms of  u,  say,  as a ratio of two
cubic polynomials in  u, each coefficient being a sum of a few products
of four coefficients in  x1, x2, y1, y2. Then  u  is determined as the
root of a polynomial of degree  9. It's pretty ghastly to write down but
of course once it is computed, its roots may be easily determined by
well-known techniques. I don't guarantee this is the most efficient method!)

dave (sci.math.research moderator)