Improvements to algorithms for computing the Minkowski sum of 3-polytopes
A Minkowski sum is a geometric operation that is equivalent either to the vector additions of all points in two operands or to the sweeping of one operand around the profile of the other without changing the relative orientation. Applications of Minkowski sums are found in computer graphics, robotics, spatial planning, and CAD. This paper presents two algorithms for computing Minkowski sum of convex polyhedron in three space (3-polytopes). Both algorithms are improvements on current ones found in the literature. One is based on convex hulls and the other on slope diagrams. The original convex hull based Minkowski algorithm is costly, while the original slope diagram based algorithms require the operation of stereographic projection from 3D to 2D for merging the slope diagrams of the two operands. Implementation of stereographic projection is complicated which increases the computation time and reduces the accuracy of the geometric information that is needed for constructing the resultant solid. This paper reports on improvements that have been made to these two algorithms and their implementation. These improvements include using vector operations to find the interrelations between points, arcs and regions on a unit sphere for the slope diagram algorithm, and addition of a pre-sorting procedure before constructing convex hull for convex hull based Minkowski sum algorithm. With these improvements, the computation time and complexity for both algorithms have been reduced significantly, and the computational accuracy of the slope diagram algorithm has been improved. This paper also compares these two algorithms to each other and to their original counterparts. The potential for extending these algorithms to higher dimensions is briefly discussed.
minkowski sum convex hull geometric algorithms polytopes math modeling of geometric variations