Direct Orthogonal Distance to Quadratic Surfaces in 3D

Discovering the orthogonal distance to a quadratic surface is a classic geometric task in vision, modeling, and robotics. I describe a simple, efficient, and stable direct solution for the orthogonal distance (foot-point) to an arbitrary quadratic surface from a general finite 3D point. The problem is expressed as the intersection of three quadratic surfaces, two of which are derived from the requirement of orthogonality of two non-coincident planes with the tangent plane to the quadric. A sixth order single-variable polynomial is directly generated in one coordinate of the surface point. The method detects intersection points at infinity and operates smoothly across all real quadratic surface classes. The method also geometrically detects continuums of orthogonal points (i.e., from the exact center of a sphere). I discuss algorithm performance, compare it to a state-of-the-art estimator, demonstrate the algorithm on synthetic data, and describe extension to arbitrary dimension.