Symbolic methods in computer graphics and geometric modeling
Abstract
Certain restricted classes of algebraic curves and surfaces admit both parametric and implicit representations. Such dual forms are useful in computer graphics and geometric modeling since they combine the strengths of the two representations. We consider the problem of computing the rational parameterization of an implicit curve or surface in a finite precision domain. Current algorithms for this problem are based on classical algebraic geometry, and assume exact arithmetic involving algebraic numbers. After applying a careful analysis of the use of algebraic numbers in current algorithms, we develop new versions of these algorithms that are more efficient. Over a certain finite precision domain we can derive succinct algebraic and geometric error characterizations, from which we conclude that our versions of the algorithms are numerically robust. A companion problem to parameterization is the accurate display of rational parametric curves and surfaces; we show how visualizing an arbitrary and possibly multiple-sheeted parametric surface is non-trivial. Such surfaces can have pole curves in their domain, where the denominators of the parameter functions vanish, and domain base points that correspond to entire curves on the surface. These are ubiquitous problems occurring even among the natural quadrics. Ordinary display techniques based on domain sampling often fail to visualize the true shape of the curve or surface. We first develop two ways of handling infinite parameter values, by using projective domain transformations. These results are then applied to the display problem. We give algorithms for parametric curves and surfaces and discuss our implementation efforts. As an implementation vehicle we developed the graphical symbolic algebra system GANITH which allows rapid prototyping of algorithms that require a blend of symbolic computation, numerical computation, and three-dimensional graphics facilities.
Degree
Ph.D.
Advisors
Bajaj, Purdue University.
Subject Area
Computer science
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