||Geometric modeling is supported by several elegant mathematical theories that assume the ability to compute and store exactly. In contrast, all practical implementations rely on real number computations with finite computational resources. This incompatibility between the theory and implementation led to emergence of new academic problems (geometric robustness and tolerant modeling) that have remained unsolved. It is also a cause of great practical difficulties with substantial economic impact that is measured in billions of dollars and created a whole new industry of geometric validation and repair. Several international standards are being created for cataloguing and categorizing both difficulties and ad hoc solutions to geometric data quality problems.
Important lessons can be learned by treating geometric models as manufactured objects, similar to traditional mechanical parts and assemblies. Implementation of the principle of interchangeability led to standardized principles of tolerancing (focused on specification and control of geometric variability) and metrology (inspection of manufactured parts through measurement and analysis of accuracy). Recent efforts focus on mathematical foundations of geometric dimensioning and tolerancing, using concepts of tolerance zones and material containments conditions.
I will argue that parallel efforts on tolerancing and metrology are needed to reconcile the theory of geometric modeling with the computational reality. It is reasonable to expect that a more general theory of geometric modeling should involve concepts of interchangeability, tolerances, and zones, but it is also important that this theory contains the classical exact theory as a special case. An important difference between tolerances in manufacturing and in geometric modeling is that manufacturing accuracy concerns exclusively with variability of real physical objects, whereas geometric errors can easily render a computer geometric model non-physical, and therefore invalid. Establishing validity conditions for a toleranced geometric model is one of the key problems in metrology of geometric models. The beginnings of the new theory are already in place, but many open problems remain.
Vadim Shapiro is a Professor of Mechanical Engineering and Computer Sciences at the
University of Wisconsin, Madison, where he founded and directs Spatial Automation Laboratory
( http://sal-cnc.me.wisc.edu/) Prior to joining the faculty of UW in 1994, he was a member of research
staff at the General Motors Research Laboratories where he worked on problems in geometric modeling,
computer aided design and manufacturing, and design automation. Shapiro received PhD degree in Mechanical Engineering
from Cornell University, MS degrees from Cornell (ME) and UCLA (CS), and BA (Mathematics and CS) from
New York University. For additional details, please visit http://homepages.cae.wisc.edu/~vshapiro/