5.6.1.2 3-D space curve

The intersection problem of a 3-D rational polynomial parametric curve and a 3-D implicit curve is defined as:

If we denote the total degree of implicit algebraic surfaces and as , and substituting , and into the implicit forms and and multiplying by , we obtain two univariate nonlinear polynomial equations and . One way to solve this problem is to compute the resultant of , , where all the coefficients of the two polynomials are known. If the resultant , then there is a common root between the two polynomials and hence we can use the inversion algorithm to find .

A robust way to solve this overconstrained problem (2 equations with 1 unknown) is to use the IPP algorithm (see Chap. 4). In such cases, the substitution must be conducted in exact arithmetic, to maintain a pristine or guaranteed precision statement of the problem.

Alternatively one could directly solve the overconstrained five-equation system in four variables ( , , , )

using the IPP algorithm as in Chap. 4.