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

曲面建模和实体建模领域一直在独立发展，它们可能构成未来计算机辅助设计(CAD)系统的基础。如今，基于一种或另一种技术的系统的用户发现，他们同时需要两种类型的系统的功能。本研究的目的是提供一种以便于将自由曲面合并到实体造型系统中的方式对自由曲面进行造型的技术。该技术中使用的曲面将由定义在平行六面体内的代数函数来描述。通过使用包含在平行六面体中的控制点，可以用B-Spline来表示代数函数，从而自动地允许整个曲面的连续性。由于隐式曲面方程定义了特定的半空间，因此这些代数曲面可以很容易地在实体建模器中实现。通过在平行六面体上剪裁它们，然后将其渲染为实体，将获得有用的自由形式基本体。光线追踪方法将有效地应用于这些曲面，因为光线相交方程是一元多项式，其次数是曲面的次数。

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.