Discretization of three-dimensional objects: approximation and convergence

In this paper, we employ a polyhedron whose vertices are only lattice points as a discrete representation of any 3D object, in order to treat the shape in a lattice space. We present a method for generating such polyhedra corresponding to the original objects in Euclidean space, and call this process discretization. Moreover, we prove that our polyhedra converge to the original objects when the grid interval is infinitely decreased to zero. The proof implies that our discretization method has the guarantee of the shape approximation for the sufficiently small grid interval. Finally, we investigate the maximum grid interval which guarantees the shape approximation.