This project investigates collision detection between two bodies based on closed-form contact space parameterization. It starts from two ellipsoids, then generalized to two convex bodies with smooth boundaries. Using nonlinear optimization techniques, the distance between the contact space and the center of one body are computed. Then, the proximity distance or penetration depth can be derived. The proposed framework is able to deal with: (1) the static case where the contact status, separation distance or penetration depth and the corresponding witness points are resulted; and (2) the continuous case where the first time-of-impact or the minimum separation distance are computed.