Determining If Two Solid Ellipsoids Intersect

Abstract

An analytical method is presented for determining if two ellipsoids share the same volume. The formulation involves adding an extra dimension to the solution space and examining eigenvalues that are associated with degenerate quadric surfaces. The eigenvalue behavior is characterized and then demonstrated with an example. The same method is also used to determine if two ellipsoids appear to share the same projected area based on an observer' s viewing angle. The following approach yields direct results without approximation, iteration, or any form of numerical search. It is computationally efe cient in the sense that no dimensional distortions, coordinate rotations, transformations, or eigenvector computations are needed. S the U.S. Satellite Catalog transitions from general perturba- tions to special perturbations, the positional accuracy of each space object will be readily available in the form of a covariance matrix. These covariances can be used to determine probability of collision, radio-frequency interference, and/or incidental laser illu- mination. Because the probability calculations can be computation- allyburdensome,itis desirable to prescreen candidate objectsbased on user-dee ned thresholds. Specie cally, each object can be repre- sented by a covariance-based ellipsoid and then processed to deter- mine if its uncertainty volume shares some space in common with another' s. Solid ellipsoids (or their projections) that do not intersect can be eliminated from further processing. This paper presents a simple analytical method to perform such screening. To date, all ellipsoidal prescreening methods involve numerical searches. 1 For computational efe ciency such prescreening is often reduced to spheres or keep-out boxes that have much larger vol- umes but allow for quick distance comparisons. The drawback to suchscreeningisthattheselargervolumescausemanyobjectstobe- come candidates for further (albeit unnecessary) processing. These methods result in increased downstream computational processing and/or increased operator workload to further assess potential satel- lite conjunctions. The following method adds an extra dimension to the solution space. The subset of eigenvalues that are associated with inter- secting degenerate quadric surfaces are then examined. The same method is also used to determine if two ellipsoids appear to share the same projected area based on viewing angle. The approach yields direct results without approximation, iteration, or any form of numerical search. It is computationally efe cient in the sense that no dimensional distortions, coordinate rotations, transforma- tions, or eigenvector computations are needed. This method ex- pands the two-dimensional work of Hill 2 in his formulation of degenerate conics (i.e., the characteristic matrix is singular). It also furthers his work by examining the associated eigenvalue behavior.