A hybrid system is a dynamical system that exhibits both continuous and discrete dynamic behavior. The state of a hybrid system changes either continuously by the flow described by a differential equation or discretely following some jump conditions. A canonical example of a hybrid system is the bouncing ball, imagined as a point-mass, under the influence of gravity. In this talk, we explore the solutions and algorithms to the extensions of this example in 3-dimension where the body of interest is rigid and convex in general. In particular, the solutions utilize the theory of nonsmooth Lagrangian mechanics to derive the differential equations and jump conditions, which heavily depend on the collision detection function. The proposed algorithm called Lie group variational collision integrator is developed using the combination of techniques and knowledge from variational collision integrators and Lie group variational integrators. If time permits, we can discuss how some of these techniques can be used in optimal control and robotics.