This is a collaborative research project between the Universities of Utah and North Carolina at Chapel Hill. The project is advancing the state of the art in geometric modeling by constructing a set of new surface representations that are sufficiently general to solve a wide range of problems in geometric surface design, estimation, and analysis. This work is founded on the assertion that no single representation is adequate to solve efficiently and accurately the myriad computational and analytical problems arising in modern applications. The strategy, therefore, is to develop a framework that systematically combines parametric and implicit surface representations. This project will examine the complementary nature of two particular representations: non-uniform B-splines (NURBs) and level sets. It will develop methods which provide: mutual tracking of the representations, analysis of the topology and singularities which occur, and mechanisms for communicating important topological events between representations. This project will also examine applications of this framework to computer aided design and terrain modeling and analysis.
When modeling 3D objects with surfaces, one is faced with a wide range of different surface representations. Each representation has corresponding drawbacks and advantages. Many modeling applications, such as computer-aided design and terrain analysis, require shapes to be modified in order to match some input data or to suit the needs of a user or designer. Therefore, one of the most important aspects of a surface representation is how it affords a user or a computer algorithm the facility to modify a shape, i.e. to deform one shape into another slightly different shape. Some representations provide a user with a great deal of control, but are constrained to a limited specific set of shapes. Other representations can represent a wide range of shapes, but only with a limited accuracy. An important concern in modifying a shape is how the representation handles cases where the surface folds back and touches itself. This is called a singularity and it indicates that the surface might be transitioning from one class of shapes to another. Different surface representations handle such singularities in different ways. This work will try to improve the technology for modifying shapes and handling singularities by combining two different shape representations and extending relevant singularity theory. This exploration should yield new tools for designing shapes and new tools for building and analyzing the shapes of measured surfaces, such as those generated from terrains. The collaboration between Utah (031075) and the University of North Carolina Chapel Hill (0310546) provides this project with specialized skills in singularity theory and unique educational opportunities for computer science students to learn more about this important subject in mathematics.
