参数曲线曲面的几何性质研究不仅对计算几何与计算机辅助几何的理论发展有重要意义，而且对曲线曲面造型技术在计算机与工程科学中的应用有重要的指导意义。基于计算几何、计算机辅助几何设计、代数几何与组合学的相关知识，本项目拟开展对一些尚未明确的参数曲线曲面几何性质进行研究，主要研究内容包括：toric曲面的升阶性质、de Casteljau算法与保凸性，toric曲面的几何连续性与分片连续toric曲面重构，所有权因子趋于无穷时NURBS曲线曲面的极限曲线曲面，NURBS曲面、有理Bezier与NURBS实体的单值性条件，有理Bezier曲线曲面、toric曲面与NURBS曲线曲面的正则控制曲面个数估计等。通过本项目的研究，促进计算几何与计算机辅助几何设计理论研究的发展，丰富曲线曲面的几何性质与算法，拓展曲线曲面造型技术在相关学科中的应用。

The study on geometric properties of parametric curves and surfaces are not only important for the improvement of the theory of computational geometry and computer aided geometric design (CAGD), but also very important for applications of curve and surface modeling in related fields of computer and engineering. This research project aims study on the geometric properties of parametric curves and surfaces, based on the theory from algebraic geometry, combinatorics, computational geometry, and CAGD. This project involves research in the following topics: the degree elevation algorithm, de Casteljau algorithm and convexity preserving of toric surface patches; the conditions for geometric continuity of toric surface patches and reconstruction of piecewise toric surface patches; the limits of NURBS curves and surfaces while all of the weights tend to infinity; the injectivity conditions of NURBS surfaces, rational Bezier and NURBS solids; the number of regular control surfaces of rational Bezier curves and surfaces, toric surface patches, and NURBS curves and surfaces. We believe the results of this project will not only enrich and improve the development of theory of computational geometry and CAGD, but also extend the applications of curve and surface modeling in related fields.