Minimizing the total projection of a set of vectors, with applications to Layered Manufacturing

Abstract

In Layered manufacturing, a three-dimensional polyhedral solid is built as a stack of two-dimensional slices. Each slice (a polygon) is built by filling its interior with a sequence of parallel line segments (of some small non-zero width), in a process called hatching. A critical step in hatching is choosing a direction which minimizes the number of segments. In this paper, this problem is approximated as the problem of finding a direction which minimizes the total projected length of a certain set of vectors. Efficient algorithms are proposed for the latter problem, using techniques from computational geometry. Experimental and theoretical analyses show that this approach yields results that approximate closely the optimal solution to the hatching problem of finding a direction which minimizes the total projected length of a certain set of vectors. Efficient algorithms are proposed for the latter problem, using techniques from computational geometry. Experimental and theoretical analyses show that this approach yields results that approximate closely the optimal solution to the hatching problem. Extensions of these results to several related problems are also discussed.