The closest and farthest points to an affine ellipse or ellipsoid

Affine ellipses/ellipsoids based bounding volumes are widely used in various graphics applications, such as ray tracing and collision detection. They provide a much tighter fit than the regular ellipses/ellipsoids. The most important operation involved is to compute the closest/farthest point, on a given ellipse/ellipsoid, with respect to a user specified point. In this paper, we first formulate such a problem for the ellipse case into solving a quartic equation and then for the ellipsoid case by solving a system of quartic equations. The method proposed in this paper is elegant and highly efficient.