Boolean operations, Minkowski sum evaluation, configuration space computation, and motion planning are fundamental problems in solid modeling and robotics. Their applications include computer-aided design, numerically-controlled machining, tolerance verification, packing, assembly planning, and dynamic simulation. Prior algorithms for solving these problems can be classified into exact and approximate approaches. The exact approaches are difficult to implement and are prone to robustness problems. Current approximate approaches may not solve these problems accurately. Our work aims to bridge this gap between exact and approximate approaches. We present a sampling-based approach to solve these geometric problems. Our approach