《代数与几何前言》翻译

Before these methods are discussed, some background remarks on representations are needed. Two primary approaches to the representation of quadric surfaces have evolved; an algebraic one and a geometric one [4]. The algebraic approach is summarized in Section 2 and is characterized by the representation of all quadric surfaces in a single form. A single surface-surface intersection algorithm suffices in this approach. The geometric approach contrasts with the algebraic one primarily in that surfaces are type-dependent combinations of scalars, points, and vectors, and algorithms for surface-surface intersections are dependent upon the types of surfaces involved[4, 7,

12].

A number of problems exist with the exclusive use of the algebraic approach. These are well documented [4, 7, 15]. Indeed, the discovery of these problems in practice led to the development and use of the geometric approach. The problems relate primarily to a lack of numerical robustness, and we amplify on some of the in Section 2 after we have developed some requisite background material.

Although geometric approaches work well when conic sections arise [5, 12], adequate methods based on these approaches when nonplanar intersecton curves result have not been described in the literature. Therefore, it has been suggested that geometric approaches be used to detect and describe conic sections when they arise, and that algebraic ones be used only after it has been determined that a nonplanar curve will result [12]. In this paper we describe how geometric approaches can be used for nonplanar intersections as well, and we note several advantages that arise from using these approaches.

We consider here only the so-called natural quadrics[7], that is, the sphere, cylinder, cone, and plane. These are by far the most commonly occurring quadric surfaces used in modeling mechanical objects. The methods described herein can be employed with many of the remaining quadrics as well. As we observe later, however, some additional techniques will be needed for some of them, and there may well come a point at which a purely geometric approach ceases to be practical or even possible.

在讨论上述方法之前，有必要提及一些关于表示方法的背景知识。数学史至今已发展出两种表示二次曲面的主要方法：代数与几何【4】，代数方法在第二节有所总结，其特点是所有二次曲面均可由一种单一形式表示，在这种方法下，一个单一的面面相交算法就足够了。而几何方法正好相反，主要在于面的表示依赖于标量，点和向量的组合，并且面面相交的算法依面的类型而定【4,7,12】。

在单独使用代数方法时出现了一些问题，这些都有详细记录【4,7,15】。实际上，正是这些在实践过程中发现的问题引领了几何方法的发展和使用。这些问题主要是缺乏数值的稳定性，在取得了一些必备的背景材料后我们在第二节对这些问题的一部分进行细致分析。

尽管几何方法在圆锥曲面出现的场合中表现很好【5，12】，但当遇到非平面相交曲线时，基于这一方法的手段在这一文献中并不充分。因此，有建议说，在圆锥截面情况下，应当使用几何方法检测和描述它们，除此之外，只有当预知结果会是非平面曲线时，才应该使用代数方法【12】。在本文中，我们也会描述几何方法是如何解决非平面相交问题的，并且，我们强调了使用这些方法的一些优点。

本文中，我们只考虑所谓的自然二次曲面【7】，也就是球面、圆柱、圆锥和平面。这些

是目前为止在机械物体建模中最常见的二次曲面。这里描述的方法也可应用于很多其他类型的二次曲面。然而，我们后来发现，这些曲面中有一些需要引用额外手法，此外，有可能出

现单纯的几何方法不再实用甚至失效的特殊情况。