5.6.3 Rational polynomial parametric/procedural parametric and procedural parametric/procedural parametric curve intersections (Cases D2 and D5)

These intersection problems are defined as:

where we have 3 equations with 2 unknowns and . Unlike the RPP/RPP curve intersection, there is no known and easily computable convex box decreasing arbitrarily with subdivision for PP curves.

A possible approach is to minimize the squared distance function between RPP curve and PP curve or between two PP curves

using numerical techniques [69]. Initial approximation may be obtained by using linear approximations of and . In general there is no guarantee to find all the stationary points.

For offsets and evolutes of rational polynomial curves, we are able to avoid the square roots of polynomials by using the auxiliary variable method, described in Sect. 4.5, so that we can apply the IPP algorithm to enhance robustness.