COMPUTATIONAL GEOMETRY FOR CAD/CAM

[Tou] Toussaint, G., " A hierarchy of simple polygons, " manuscript in preparation. Determining the castability of simple polyhedra, " manuscript in preparation. Mould design with sweep operations-a heuristic search approach, " C.In the manufacturing industry, developing machines that perform orthogonal and opposite cast removal is much simpler than producing machines that perform arbitrary cast removal. In fact, opposite cast removal seems to be the most popular technique used ([Pri87], [PC93]). Furthermore, if orthogonal or opposite cast removal is possible, it can be determined more efficiently. We summarize the complexity of the different algorithms [BBV93] developed for the casting problem. orthogonal opposite arbitrary convex polyhedra O(n log 2 n) O(n log 2 n) O(n 2 log n) simple polyhedra O(n 2) O(n 2) O(n 2 log n) 5 Stereolithography In this section, we consider the problem of deciding whether or not a design is feasible for a CAD/CAM system developed and patented by 3D Systems of Sylmar, CA that employs a process called stereolithography. For details the reader is referred to [ABB93]. The components of the stereolithography manufacturing process consist of a vat of liquid photocur-able plastic, a computer controlled table T on a stand S that can be moved up and down in the vat and a laser L above the vat that can shine on the surface of the liquid plastic and can move in a horizontal plane. The system works as follows. At the first step the table is just below the surface of the plastic and the laser is controlled to move about so that the light shines on the surface of the plastic and draws the bottom-most cross-section of the object A being built. When the laser light contacts the plastic, the plastic solidifies and so the first cross-section of the object is formed and rests on the table. At the next step the table is lowered a small amount to allow liquid to cover the hardened layer and the laser then draws the next cross-section of the object. The light from the laser penetrates the liquid just deep enough so that this cross-section is welded to the lower cross-section produced at the previous step. This process is repeated until the entire object is formed. The direction given by a normal to the table pointing to the laser is called the direction of formation for the object. There are some objects that can be formed …