# ICS 269, Winter 1997: Theory Seminar

## 14 Feb 1997:

Surface/Surface Intersection '97

Mac Casale, ICS, UC Irvine

In high-school algebra, it is common to learn how to intersect
two geometric objects (starting with straight lines) that are
described by algebraic equations (e.g., *y* = *
mx* + *b*). In freshman calculus, the concept is
extended to vector functions of a parameter. In both courses, some
kind of substitution method is usually used. If a student goes
further, he/she finds that substition methods had been extended
long ago, under the name of elimination theory, to a high level of
sophistication. (See the famous *Theory of Equations*, by J.
V. Uspensky, McGraw Hill 1948). It may therefore come as a surprise
that surface/surface intersection is still undergoing considerable
research in the domain of Computer Aided Geometric Design.

In this paper, we will present the progress in algorithms that
have been applied to intersecting two surfaces embedded in **
R**^{3}, with varying degree of success. It will be shown
that this algorithm, which is central to the robustness of modern
CAD systems, is inherently hard, not from the point of view time or
space complexity but due to

- The chaotic nature of the requisite non-linear numerical
methods, and
- The rich topology of intersection curves among intersecting
surfaces that occur in real world problems.

Practical, state-of-the-art methods for overcoming these
difficulties will be presented.