Use of Free-Form Low-Degree Algebraic Surfaces in Solid Modeling

曲面建模和实体建模是未来计算机辅助设计（CAD）系统的基础，它们正在独立发展。今天，基于一种或另一种技术的系统用户发现他们同时需要两种类型系统的功能。本研究的目的是提供一种技术来模拟自由曲面，以促进其在实体建模系统中的结合。在这种技术中使用的曲面将由平行六面体内定义的代数函数来描述。通过使用平行六面体中包含的控制点，代数函数可以用b样条表示，自动允许整个表面的连续性。这些代数曲面可以很容易地在实体建模器中实现，因为隐式曲面方程定义了特定的半空间。通过对平行六面体进行裁剪，然后将其渲染为实体，可以获得有用的自由形式原语。光线追踪方法将有效地应用于这些表面，因为光线相交方程是一个单变量多项式，其程度是表面的程度。

The fields of surface modeling and solid modeling, which are likely to form the basis for future computer-aided design (CAD) systems, have been developing independently. Today users of systems based on one or the other technique find they require the functionality of both types of systems simultaneously. The goal of this research is to provide a technique for modeling free-form surfaces in a manner which facilitates their incorporation in solid modeling systems. The surfaces used in this technique will be described by algebraic functions defined within a parallelepiped. By using control points contained in the parallelepiped, the algebraic function can be expressed with B-splines, automatically allowing for continuity throughout the surface. These algebraic surfaces can be implemented easily in solid modelers since the implicit surface equation defines specific half- spaces. A useful free-form primitive will be obtained by clipping them against the parallelepiped and then rendering as solids. Ray tracing methods will be efficiently applied to these surfaces, since the ray intersection equation is a univariate polynomial whose degree is the degree of the surface.