EXPLICIT SOLUTION FOR THE MINIMUM DISTANCE BETWEEN TWO SOLID SEMI-INFINITE CIRCULAR CONES

Multi-body kinematics and object rendering often involve minimum distance calculations. Explicit solutions exist for the distance between spheres, cylinders and other simple objects. Deriving the minimum distance between cones requires numerical minimization or geometrical approximations combined with analytical solutions for the simpler objects. This paper describes an explicit solution for the minimum distance between two solid semi-infinite circular cones. The method combines geometrical reasoning with analytical derivation. The solution also includes the location of the intersection points. Solution regions are identified and discussed. A numerical method based on minimizing the distance between two cone generators was used as part of the verification process. The exact solution was compared to results of approximation by regular polytopes. The explicit solution is robust, independent of coordinate system and invariant under rigid translation and rotation of the setup.