5.8.3 Implicit algebraic/implicit algebraic surface intersection (Case F8)

Implicit algebraic surface to implicit algebraic surface intersection is defined as follows:

where are polynomial functions. Here we have two equations in three unknowns . Bajaj et al. [17] developed a marching method for IA/IA surface intersection as well as for parametric surfaces.

A method for low order , is to eliminate one variable (e.g. ) to find projection of intersection curves on the plane of other two variables (e.g. , ), then trace the algebraic curve and use the inversion algorithm to find . Intersections of low degree implicit algebraic surfaces are of special interest in the boundary evaluation of the Constructive Solid Geometry models. A more complete analysis of the special intersections of two quadric surfaces (used frequently in CAD/CAM of mechanical parts) can be found in [233,234,367,104,443,390,268].

Example 5.8.4. Consider the intersection of a sphere and a circular cylinder given by

as shown in Fig. 5.26. The projection of the intersection curves on the three coordinate planes is illustrated in Fig. 5.27.

Hartmann [155] proposed the idea of numerical implicitization which allows treatment of intersection problems of not only parametric surfaces but also non-standard surfaces such as an offset of an implicit surface, a Voronoi surface, an envelope of a one parametric family of spheres etc. The key idea is that in tracing the intersection curve of two implicit surfaces, we are only required to calculate the implicit function values and the gradients of the implicit functions at the intersection points as in Bajaj et al. [17]. In other words, we do not need to know the functions explicitly. Therefore if we can implicitize any two surfaces numerically we are able to trace the intersection curve using the IA/IA surface intersection algorithm of Bajaj et al. [17].

Figure 5.26: Intersection of two implicit quadrics (sphere and cylinder)

Figure 5.27: (a) Projection of intersection curve on ( ) plane, i.e. , (b) projection of intersection curve on ( ) plane, i.e. , (c) projection of intersection curve on ( ) plane, i.e.