The approach for the combined methods is to combine the rational parametric description of one surface p₁(s,t) , with the algebraic representation of the other surface q₂(x,y,z)=0. Thus the problem is converted to a problem of finding the topology of an algebraic curve f(s,t)=q₂(p₁(s,t))=0 in the parametrization of the first surface.

A limited number of critical points

 The approach is based on finding critical point, points where either

• ∇f(s,t)=0
• ∂f(s,t)/∂(s)=0

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.

Projection to first parameter direction

The problem is project to a polynomial in the first parameter varaible of f(s,t) by:

• Computing the discriminant R(s ) of f(s,t) with respect to t, and finding the real root of  R(s), α₁,...,α_r. Sturm-Habicht sequences here supplies an exact number of real root in the interval of interest.
• Then for each α_i, i=1,...,r  we compute the real roots of f(α_i,t), β_i,j, j=1,...,s_i,.
• For every α_i and β_i,j  compute the number of half brancehsto the right and left of the point (α_i,β_i,j).

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. 

Reconstruction of topology of the algebraic curve

From the above information the topology of the algebraic curve in the domain of interest can be constructed.

Software

The algorithms have been implemented using symbolic packages.