Rational polynomial parametric surface to rational polynomial parametric
surface intersection is defined as:
(5.99)
Formulation can be provided by setting
which leads to three
nonlinear polynomial equations for four unknowns
. It
is an underconstrained system with 3 equations and 4 unknowns. This
system can be solved by the IPP algorithm of Chap.
4. However, as the solutions are typically not isolated
points but curves, such approach is very slow when small tolerances
are used. One could also implicitize
to the
form
and substitute
,
and
into
to reduce the problem to Case F3 for low degree surfaces
[212]. Heo et al. [160] studied the intersection
of two ruled surfaces which is simpler than the general parametric
surface to surface intersection problem.
There are three major techniques for solving RPP/RPP surface
intersections. Detailed reviews can be found in
[295,300].