Extension of current intersection methods applied on rational B-spline surfaces, to more general and complex surfaces requires further study. Such surfaces include offset, generalized cylinder (pipe or canal surfaces in particular), blending, and medial surfaces and surfaces arising from the solution of partial differential equations or via recursion techniques (subdivision surfaces [354]). Intersections of such surfaces with the basic low order algebraic and rational B-spline surfaces, commonly used in CAD need to be explored. However, a basic element of a solution of many of these problems is the auxiliary variable method described in [169,253,300], where the problem is reduced to a higher dimension nonlinear polynomial system. In some cases, recent research has indicated that some special instances of these general surfaces can be exactly expressed as rational polynomial surfaces [324,241,256,250] of higher degree and therefore these problems are reducible at least in principle to the problems addressed in this chapter. Further research is needed to implement this idea in a practical setting and examine the relative efficiency of competing approaches.
Investigating the effects of floating point arithmetic on the implementation of intersection algorithms has been an important area for basic research during the last decade. Ways to enhance the precision of intersection computation, to monitor numerical error contamination and alternate means of performing arithmetic, not relying on imprecise floating point computation alone, have been explored in some detail. Researchers in surface intersection problems during the last decade have already obtained a good understanding of robustness problems when employing floating point arithmetic and of methods to mitigate these problems based on normalization of the system [133] and rounded interval arithmetic [178]. However, these methods are not a panacea since they cannot resolve effectively non-zero-dimensional solution sets of nonlinear systems or achieve very high precision in reasonable computation times. A related active problem area has been the rectification of solid models expressed in the Boundary Representation form, which attempts to resolve intersection inconsistencies in such models and create topologically and geometrically consistent models [303,360,388,389].
As a result of these deficiencies, recent research tends to focus on exact methods involving rational arithmetic [196,356,358]. Much research remains to be done in bringing such methods to the CAD practice, generalizing the arithmetic to go beyond rational and algebraic numbers (eg. involving transcendental numbers of trigonometric form), and to explore more efficient alternatives that are generally applicable in low and high degree problems alike. Finally, a general and comprehensive comparison and mapping of the efficiency properties of all available methods for solving nonlinear systems robustly would be valuable as a guide for future research.