Fast Numerical Operations on Implicit Curves and Surfaces

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

Strain 0209617 Robust global implicit representations of curves and surfaces simplify geometric operations such as evolving, offsetting and intersecting, while requiring accurate contouring and efficient implicitization techniques. The investigator focuses on fast new algorithms for the three key issues of implicit representations: 1. Fast accurate multiresolution methods that extract piecewise-smooth contours with tightly controlled geometry. These methods alternate between global topology resolution and local differential-algebraic boundary value problems, and solve industrial problems ranging from machine tooling to horizon finding. 2. Efficient implicitization methods that build a distributed implicit representation from given curves or surfaces. These methods, which invert contouring, are essential to modeling moving material interfaces such as the solid-liquid interface in melt growth of silicon crystals. 3. Fast stable deferred correction methods for the accurate solution of ordinary and partial differential-algebraic initial and boundary value problems that control local geometry. Differential-algebraic problems occur widely in engineering, and challenge existing numerical methods. For example, automated vehicle piloting requires real-time solution of differential systems with constraints that keep the tires on the road. Computer models of complex interlocking physical objects dominate technological processes ranging from semiconductor etching to surgery planning, from computer animation to computer-controlled machine tooling. Key steps in these processes involve operations such as evolving, navigating, intersecting and blending on these objects. The investigator and his students are developing efficient new computational tools for simulating these operations in robust user-friendly ways. This highly interdisciplinary enterprise combines mathematics and computer science to benefit scientific endeavors ranging from cartography to crystallography. The research objectives intertwine with an interdisciplinary educational program, training future scientists and engineers through web-enabled courses, math/science/engineering seminars, and individual student research mentoring.