The fundamental issue in intersection problems is the efficient
discovery and description of all features of the solution with
high precision commensurate with the tasks required from the
underlying geometric modeler [295,300].
Reliability of intersection algorithms is a basic prerequisite for
their effective use in any geometric modeling system and is closely
associated with the way features of the solution such as constrictions
(near singular or singular situations), small loops and partial
surface overlap are handled. The solutions resulting from most
present techniques, implemented in practical systems, are further
complicated by imprecisions introduced by numerical errors present in
finite precision computations.
Intersection problems can be classified
according to the dimension of the problem and according to the type
of geometric equations involved in defining the various geometric
elements (points, curves and surfaces). The solution of intersection
problems can also vary according to the number system in which the
input is expressed and the solution algorithm is implemented. Such
intersection problem classification is addressed in the next three
subsections. Only the most important intersection problems are
addressed in detail in Sects. 5.3 to
5.8.
