Feasability of Design in Stereolithography

We study the feasibility of design for a layer-deposition manufacturing process called stereolithography which works by controlling a vertical laser beam which when targeted on a photocurable liquid causes the liquid to harden. Given an object (modeled as a polygon or a polyhedron), we give algorithms that decide in O(n) time whether or not the object can be constructed using stereolithography. Furthermore, if the answer is in the affirmative, the algorithm reports a description of all the positions in space in which the object can be made. We also determine feasibility of both polygonal and polyhedral objects constructed using variable-angle stereolithography. We give an O(n) time algorithm for polygons and O(n log n) as well as O(n) time algorithms for polyhedra. We show that objects formed using variable-angle stereolithography can also be constructed using another manufacturing process known as injection molding. Finally, we show that the polyhedral objects formed by stereolithography are closely related to polyhedral terrains which are important structures in geographic information systems (GIS) and computational geometry. In fact, our algorithms recognize whether a polyhedral surface is a terrain that allows overhangs, thus initiating the study of more realistic terrains than the standard ones considered in geographic information systems.