分片代数超曲面的构造与参系数分片多项式系统的研究进展

Progress in construction of piecewise algebraic hypersurfaces and solving parametric piecewise polynomial systems

多元样条是具有一定光滑度的分片多项式,具有一定光滑度的分片代数(超)曲面(即多元样条的零点集)是表示或逼近曲面的重要工具.研究一种有效方法用于构造具有一定光滑度和预先给定拓扑的实分片代数超曲面是解决如何表示或逼近具有一定拓扑结构(特别是复杂拓扑结构)的几何物体问题的重要途径之一,也是计算几何与代数几何研究中一个新的重要主题.参系数分片多项式系统不仅与曲面相交、拼接和过渡曲面生成等一系列研究密切相关,而且是参系数半代数系统的本质推广.本文介绍分片代数超曲面的构造与参系数分片多项式系统的一些最近的研究进展.

The multivariate spline is a piecewise polynomial with certain smoothness, and the pmcewlse alge- braic hypersurfaces with certain smoothness （i.e. the set of all common zeros of multivariate splines） have become useful tools for representing or approximating surfaces. Studying an effective method for the construction of real piecewise algebraic hypersurfaces of a given degree with certain smoothness and prescribed topology is one of the important ways to solve the problem of how to represent or approximate geometric objects with certain topological structure （especially the complex topology structure）, and also a new and important topic of compu- tational geometry and algebraic geometry. Parametric piecewise polynomial systems are not only closely related to a series of reasearch, such as intersection of surfaces; blending curves and surfaces; generation of transition surfaces, but Mso essentiM generalization of the parametric semi-Mgebraic systems. The purpose of this paper is to introduce some recent research progress on the construction of real piecewise algebraic hypersurfaces, and parametric piecewise polynomial systems.