Simple but not Simpler: A Surface-Sliding Method for Finding the Minimum Distance between Two Ellipsoids

Abstract

We propose a novel iterative process to establish the minimum separation between two ellipsoids. The method maintains one point on each surface and updates their locations in the theta-phi parametric space. The tension along the connecting segment between the two surface points serves as the guidance for the sliding direction, and the distance between them decreases gradually. The minimum distance is established when the connecting segment becomes perpendicular to the ellipsoid surfaces, at which point the net effect of the segment tension disappears and the surface points no longer move. Demonstration examples are carefully designed, and excellent numerical performance is observed, including accuracy, consistency, stability, and robustness. Furthermore, compared to other existing techniques, this surface-sliding approach has several attractive features, such as clear geometric representation, concise formulation, a simple algorithm, and the potential to be extended straightforwardly to other situations. This method is expected to be useful for future studies in computer graphics, engineering design, material modeling, and scientific simulations.

Figures (7)

• Figure 1: A graphic representation for the notations adopted in this article for an ellipsoid.
• Figure 2: A graphic representation of the search process for the minimum distance between two ellipsoids, with key vectors used in the calculation illustrated.
• Figure 3: The search process for System I in Table \ref{['tbl:parameters']}: (a) a 3D view of the relative positions and orientations of the two ellipsoids, (b) the evolution of the surface point locations, (c) the converging process in distance between the surface points, (d) the relative error in separation distance, (e) the refining process of the angle increment steps, and (f) the improving alignment between the surface normal directions and the connecting segment. In (a), the circles represent the initial surface point positions and the squares for their final locations when the minimum distance is found, and the curves over the ellipsoid surfaces are the surface point trajectories.
• Figure 4: Influences of (a) different initial angle increment step $\lambda^0$ and (b and c) different initial positions of the surface points. In (c), the circles represent the initial surface point positions at the beginning of the search, the squares represent their final positions when the solution is found, and the curves between them indicate the search paths in the $\theta-\phi$ plane. Different colors are used: blue for ellipsoid $\mathcal{E}_1$ and red for ellipsoid $\mathcal{E}_2$. The same line styles are adopted in (b) and (c) for each case for clarity.
• Figure 5: 3D views for System II in Table \ref{['tbl:parameters']} (a for the aligned configuration and b for the rotated configuration). The circles represent the initial surface point positions and the squares for their final locations when the minimum distance is found, and the curves over the ellipsoid surfaces are the surface point trajectories. The search processes with these two configurations are compared in (c) for the separation distance $d$ and (d) for the search paths in the $\theta-\phi$ plane.