Previously, we have describe modal analysis, an efficient, physically-based solution for recovering, tracking, and recognizing solid models from 2-D and 3-D sensor data. The underlying representation consists of two levels: modal deformations, which describe the overall shape of a solid, and displacement maps, which provide local and fine surface detail. In this paper, we give details about the mathematics behind implicit function and displacement map calculations. In addition, we describe an extension which can be used to incorporate measurement uncertainty in the recovered modal deformation parameters. The result is an energy-based implicit function; as a consequence, collision detection, path planning, dynamic simulation, and model comparisons can frequently be performed in closed-form--even for complex shapes.