A surface of revolution with moving axes and angles is a rational tensor product surface generated from two rational space curves by rotating one curve (the directrix) around vectors and angles generated by the other curve (the director). Here we introduce these new kinds of rational generalized surfaces of revolution, provide some interesting examples, and investigate their algebraic and geometric properties. In particular, we study the base points and syzygies of these rational surfaces. We construct three special syzygies for a surface of revolution with moving axes and angles from a μ-basis of the directrix, and we show how to compute the implicit equation of these rational surfaces from these three special syzygies. Quaternions and quaternion multiplication are used to represent these surfaces. Examples are provided to illustrate our theorems and flesh out our algorithms.

具有移动的轴和角度的旋转表面是由两个有理空间曲线生成的有理张量积表面，通过将一个曲线（准线）围绕另一条曲线（指向矢）生成的矢量和角度旋转。在这里，我们介绍这些新型的有理广义旋转面，提供一些有趣的示例，并研究它们的代数和几何性质。特别是，我们研究了这些有理曲面的基点和同构。我们为旋转表面构造了三种特殊的syzygies，它们的轴和角度从μ起-的基础上，我们展示了如何从这三个特殊的sysygies计算这些有理曲面的隐式方程。四元数和四元数乘法用于表示这些曲面。提供示例来说明我们的定理并充实我们的算法。