This paper presents two practical methods for computing the closest approach distance of two ellipsoids in their inter-center direction. The closest approach distance is crucial for collision handling in the dynamic simulation of rigid and deformable bodies approximated with ellipsoids. To find the closest approach distance, we formulate a set of equations for two ellipsoids contacting each other externally in terms of the inter-center distance, contact point, and normal vector. The equations are solved robustly and efficiently using a hybrid of the fixed-point iteration method and bisection method with root bracketing, and a hybrid of Newton’s method and the bisection method. In addition to a stopping criterion expressed with the progress of the solution, we introduce a novel criterion expressed in terms of the error in distance. This criterion can be effectively employed in real-time applications such as computer games by allowing an unnoticeable error. Experimental results demonstrate the robustness and efficiency of the proposed methods in various experiments.

中文翻译：

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计算两个椭球的最近距离

本文介绍了计算两个椭球在其中心方向上的最近接近距离的两种实用方法。最近接近距离对于近似椭球体的刚性和可变形体的动态模拟中的碰撞处理至关重要。为了找到最近的接近距离，我们根据中心间距离、接触点和法向量为两个在外部相互接触的椭球制定了一组方程。使用定点迭代法和带根括号的二分法的混合方法，以及牛顿法和二分法的混合方法，可以稳健有效地求解方程。除了用解决方案的进度表示的停止标准之外，我们还引入了一种用距离误差表示的新标准。通过允许不明显的错误，该标准可以有效地用于实时应用程序，例如计算机游戏。实验结果证明了所提出的方法在各种实验中的鲁棒性和效率。