RIA: Cyclides as Blending Surfaces in Geometric Modeling

9410707使用几何方法，该项目旨在彻底解决将Dupin Cyclde用作所有CSG基元对的混合曲面的问题，并探索其他易处理的曲面，例如Cyclde族，可以用于混合目的的可能性。最初，该项目将研究Dupin Cycldes作为CSG基本体的混合曲面的作用。这包括每对CSG基元及其泛化的构造算法和表征结果。由于构造是基于几何特征的，而不是基于隐式/参数方程，因此构造算法将更加健壮。接下来，我们将考察Dupin环线的各种推广(例如，射影环线、一般环线和具有双二次曲线的四次曲线)，以确定是否存在可行且易于处理的低次代数曲面用于混合目的。画法几何、代数几何和微分几何的技术将被用来研究这些曲面类的几何和开发有效的构造算法。最后，通过观察相交曲线、剪裁曲线和过渡曲面的几何之间的联系，来考虑一般过渡曲面构造的一些基本问题。***

9410707 Shene Using a geometric approach, this project aims to completely solve the problem of using Dupin cyclides as blending surfaces for all pairs of CSG primitives, and to explore the possibility that other tractable surfaces, such as the cyclide family, can be used for blending purpose. Initially, the project will investigate the role of Dupin cyclides as blending surfaces for CSG primitives. This includes construction algorithms and characterization results for each pair of CSG primitives and their generalizations. Since constructions are based on the geometric characteristics, not on implicit/parametric equations, the construction algorithms would be more robust. Next, various generalizations of Dupin cyclides (e.g., projective cyclides, general cyclides and quartics with a double conics) will be examined to see if there are viable and tractable classes of lower degree algebraic surfaces for blending purpose. Techniques from descriptive, algebraic and differential geometry will be used to study the geometry of these surface classes and to develop efficient construction algorithms. Finally, some fundamental issues of general blending surface construction will be considered by looking at the connection between the geometry of the intersection curve, the clipping curves, and the blending surfaces. ***