A Practical Method to Compute the Closest Approach Distance of Two Ellipsoids

두 타원체 사이의 최단 근접 거리를 구하는 실용적인 방법

Abstract

This paper presents a practical method to compute the closest approach distance of two ellipsoids in their inter-center direction. This is the key technique for collision handling in the dynamic simulation of rigid and deformable bodies approximated with ellipsoids. We formulate a set of equations with the inter-center distance and the contact point and normal for the two ellipsoids contacting each other externally. The equations are solved using fixed-point iteration and Aitken's delta-squared process. In addition, we introduce a novel stopping criterion expressed in terms of the error in distance. We demonstrate the efficiency and practicality of our method in various experiments.

본 논문에서는 두 타원체 사이의 중심 간 방향으로의 최단 근접 거리를 구하는 실용적인 방법을 제안한다. 이는 타원체로 근사한 강체 및 변형체의 물리기반 동적 시뮬레이션에서 타원체 사이의 충돌을 처리 하는 핵심 기술이다. 본 논문에서는 외부에서 접하는 두 타원체의 중심 간 거리와 접촉점 및 접촉방향에 관한 조건식을 세우고 고정점 반복법 및 Aitken의 델타 자승 절차를 이용하여 최단 근접 거리를 구하는 방법을 개발한다. 또한 실제 오차에 따른 종료 조건을 도입함으로써 게임 등의 실시간 응용에서 최단 근접 거리를 더욱 빠르게 구할 수 있게 한다. 다양한 실험을 통해 제안된 방법의 효율성 및 실용성을 보인다.

[Fig. 1] Two contacting ellipsoids e₁(x) = 1 and e₂(x-dn) = 1 with the closest approach distance d in the inter-center direction n and their concentric ellipsoids (with the same aspect ratios) contacting along the curve x(λ) where the gradient vectors ∇e₁(x) and ∇e₂(x-dn) are parallel.

[Fig. 2] Distribution of the number of solver iterations required to satisfy |_λi+1 - λ_i| < 10^-8 for 10 million pairs of ellipsoids generated randomly with γ = 3 and Γ = 3.

[Fig. 3] Average and maximum numbers of solver iterations for one million samples generated for each γ increasing from 1 to 10 by 0.1.

[Fig. 4] Average and maximum numbers of solver iterations for one million samples generated for each Γ increasing from 1 to 100 by 1.

[Fig. 5] Average and maximum numbers of solver iterations for one million samples when Ε of the stopping criterion |λ_i+1 - λ_i| < Ε decreases from 1 to 10-10 by 10-1.

[Fig. 6] Average and maximum numbers of solver iterations for one million samples when $\epsilon$ of the stopping criterion ${\mid}{\mid}x_1^{(i)}-x_2^{(i)}{\mid}{\mid}$ < ${\epsilon}r_{min}$ decreases from 1 to 10-10 by 10-1.

[Fig. 7] Real-time simulation of 180 deformable models with 4,404 ellipsoidal particles using the as-rigid-as-possible solid simulation technique presented in [3].

[Fig. 8] Distribution of the number of solver iterations for 49,716,944 pairs of ellipsoids with |λ_i+1 - λ_i| < 10^-8.

[Fig. 9] Distribution of the number of solver iterations for 49,716,944 pairs of ellipsoids with ${\mid}{\mid}x_1^{(i)}-x_2^{(i)}{\mid}{\mid} < 10^{-2}r_{min}$.