The approach for the combined methods is to combine the rational parametric description of one surface p_{1}(s,t) , with the algebraic representation of the other surface q_{2}(x,y,z)=0. Thus the problem is converted to a problem of finding the topology of an algebraic curve f(s,t)=q_{2}(p_{1}(s,t))=0 in the parametrization of the first surface.
The approach is based on finding critical point, points where either
For any value in the first parameter direction of f(s,t) there will be a limited numer of such critical points. There are also a finit number of rotations of f(s,t) that will have have more than one critical point. f(s,t) is rotated to ensure that for a given value there will be only one critical point.
The problem is project to a polynomial in the first parameter varaible of f(s,t) by:
To ensure the approach to work the root computation has to use extended precision to ensure that we reproduce the number of roots predicted by the Sturm-Habicht sequences.
From the above information the topology of the algebraic curve in the domain of interest can be constructed.
The algorithms have been implemented using symbolic packages.
Published June 7, 2005