Presidential Young Investigator Award: Loop Detection in Surface Patch Intersections

这位总统青年研究员在几何建模方面的研究集中在四个领域：曲面片相交、三次代数曲面片、局部曲面隐含化和代数曲线的近似参数化。对于曲面片相交，他计划继续计算两个参数曲面片的相交曲线。他已经找到了一种方法来确定相交曲线中是否有任何闭合环，并计划开发一种利用这种环测试的通用相交算法。在三次代数曲面片中，他计划研究用三阶代数曲面片建模的方法。前人的工作证明了它们可以以C_1连续性拼接在一起，并且可以对它们施加有理的参数化。他正在致力于创造计算机环境来对它们进行建模。局部曲面隐含化的问题包括找出参数曲面片与低次隐式曲面的逼近程度，并开发一种计算此类曲面的算法。例如，一个双三次曲面通常有一个18次的隐式方程，但人们希望大多数双三次曲面都可以用四次或五次曲面来逼近。最后，对于任何参数曲线，都可以找到精确的隐式方程，但反之亦然。塞德伯格教授正在开发一种算法，将分段有理三次曲线与代数曲线进行拟合。

This Presidential Young Investigator's research in geometric modeling focuses on four areas: surface patch intersections, cubic algebraic surface patches, local surface implicitization, and approximate parametrization of algebraic curves. For surface patch intersections, he plans to continue work on computing the curve of intersection of two parametric surface patches. He has already found a method to determine if there are any closed loops in an intersection curve and plans to develop a general intersection algorithm which takes advantage of this loop test. In cubic algebraic surface patches, he plans to investigate methods of modelling with third order algebraic surface patches. Previous work proves that they can be pieced together with C1 continuity and that a rational parametrization can be imposed on them. He is working on creating the computer environment for modelling them. The problem of local surface implicitization involves finding how closely a parametric surface patch can be approximated by an implicit surface of low degree and developing an algorithm for computing such surfaces. For example, a bicubic patch generally has an implicit equation of degree 18, but it is hoped that most bicubic patches can be approximated with a degree four or five surface. Finally, an exact implicit equation can be found for any parametric curve, but the reverse is not true. Professor Sederberg is working on developing an algorithm for fitting a piecewise rational cubic curve to an algebraic curve.