A fast numerical procedure is presented for computing growth distances between a pair of polytopes in three dimensional space that move incrementally along specified smooth paths. The procedure carrys out the growth distance evaluations efficiently by predicting and verifying contact configurations between a pair of grown polytopes. In the prediction and verification the procedure uses vertex and facial characterizations of polytopes and exploits their geometric adjacency information. The computation time, in average, is very small and does not depend significantly on the geometric complexity of two polytopes. A numerical example is presented to demonstrate the applicability of the procedure to interference detection in robotic simulations.