椭圆抛物面/球面求交算法

摘要

基于广义特征多项式给出了判断椭圆抛物面与球面是否有交的方法,在有交的条件下,得到了交线为圆的条件以及圆心、半径和法向量等重要几何参数,确保了交线的准确绘制.以平行圆族为基础进行坐标变换,在新坐标系下将椭圆抛物面的一个参数方程代入球面的方程中,得到一个一元四次方程,由方程根的分布情况确定交线的拓扑结构,在每一个有交子区间上,给出所有交曲线段的参数方程.给出了几个具体的实例进行说明.%Based on the generalized characteristic polynomial, the method for judging whether an elliptic paraboloid and a sphere have intersection curves is presented.The condition that intersection curve is a circle and the important geometric parameters such as circle center,radius and normal vector are obtained,which ensures the intersection curves are drawn accurately. A coordinate transformation based on parallel circular family is achieved.A quartic equation with one unknown is obtained by substituting the parameter equation of the elliptic paraboloid into the equation of the sphere.According to the distribution of the roots,the topological structure of the intersection curves can be determined.In those subintervals that intersection points lie in,intersection curves are provided in a parametric form.Some examples are provided to demonstrate the algorithm.