圆环面/球面求交算法

Torus/Sphere Intersection Algorithm

将圆环面看成中心在大圆上的一族圆,从而将球面圆环面求交的问题转化为球面与一族圆的求交问题.该算法不需要跟踪交线.首先利用点圆最近距离的理论,直接判断是否无交、相切于一点、交于一个圆或交于两个圆等简单的情况;其他情况下,通过求解关于圆环面大圆的参数的一元四次方程的根,然后对该参数区间[0,2π]进行划分,并通过简单的符号判断来确定有交的参数子区间,在这些有交的子区间上直接给出所有交曲线段的参数表示形式.

The torus/sphere intersection problem could be converted into the intersection problem between a sphere and a cluster of circles if a torus is considered as a cluster of circles with centers on an outer circle. No tracing is required at all. With the theory of the minimum distance between a point and a circle, some special cases are directly figured out such as no intersection, one tangent point, one intersecting circle, or two intersecting circles. For other cases, the intersection problem is solved by a quartic equation with respect to the parameter of the central circle of the given torus. The parametric interval [0, 2π] is divided and a sign-detection method is presented to find out those intervals that intersection points lie in. The resultant curves are provided in a parametric form.