| Description |
Geometric algorithms based on floating point arithmetic often fail or generate incorrect results due to floating point arithmetic errors and geometric approximation errors, Ambiguous interpretations of degenerate cases in geometric modeling, for instance, can lead to invalid geometric representations and system failures. There has been no provably robust approach to this problem.; This dissertation presents two methods for robustness in geometric computation, in general, and solid modeling, in particular. The approach taken is intuitionistic and tolerance-based, namely detecting geometric relations using tolerances of the geometric objects and dynamically updating the tolerances to preserve the properties of the geometric relations. Primary contributions of this dissertation include:; • Formal definitions of geometric robustness using the notion of representation and model and the concept of intuitionistic geometry; • Two robustness methods, the analytic model method and the approximated model method. Their respective domains of applications are geometric problems for (a) low degree analytic curves and surfaces (e.g., lines, circles and planes) and (b) high degree curves and surfaces (c.g., sculptured solid modeling).; • An ambiguity handling approach using dynamic tolerance adjustments.; Applications of the two methods in various geometric modeling problems are also discussed. |
