Designing molds in which to cast solids

Casting is a process of creating solid objects, usually metals or plastics, by pouring the material in a liquefied state into a mold. The shape of the mold is of course dependent on the shape of the object that is to be cast. A variety of solids can be cast using molds but this summary restricts its attention to objects that can be cast using only a single mold. Consider for example the case of casting a sphere. There does not exist a single mold from which a sphere can be cast. If, however, it was permitted to use more than just one mold, the sphere could be cast in two halves.

This previous example should give the reader some intuition as to the constraints that must be imposed on a castable object. Before the constraints are outlined, let us make certain assumptions about the nature of the castable object. Firstly, we consider the solid to be casted to be a polyhedron. Quite informally, a polyhedron can be defined as a three dimensional object that is bounded by a finite set of planar facets. Given this definition of a castable object, we can quite easily see that the object must have a top facet that is coplanar with the top facet of the mold. Given that the mold has its base in the XY plane, the top facet of the object must be parallel to the XY plane too.

This is the problem that is to be addressed: Given a solid polyhedron, does there exist a mold from which it can be removed? A constraint on the removal of the object from the mold is that it may be removed only with a single translation from the mold. This constraint precludes the possibility of rotations etc. which would mean that a screw, for example, is an uncastable object.

We can restate the problem in the following terms. Given a solid object in a mold, does there exist a direction, d, in which the object can be translated so as to remove it from the mold. This direction d has to be such that the object when translated by this vector makes no collisions with the interior of the mold. A little reflection will convince the reader that the direction d must be chosen such that the angle made between d and the normals to each of the facets of the mold must be greater than or equal to 90 degrees of arc.

The authors propose and prove a lemma of importance, the statement of which is equivalent to the statements in the previous paragraph. From this lemma, the corollary follows, that if an object can be removed from a mold with a series of translations, then it can be removed by a single translation. Finding this direction, d, then becomes the crux of the problem.

A Glossary of terms encountered
Coplanar: 	The property of lying in the same plane
Polyhedron: 	A 3D region of space bounded by a finite set of planar facets
Translation: 	A linear displacement specified by a single vector
Normal:		A direction that is orthogonal to a specified set of vectors
Keywords in the problem
FACET: 		A section of a plane that bounds the solid under consideration
LINEAR PROGRAMMING: The solution to the problem is found using a 
technique  of optimization known as linear programming. This method is 
typified by the existence of numerous linear constraints on a problem, 
the optimal solution of which lies at one of the vertices of the 
convex polyhedron generated by the linear constraints.


Quick Review of Linear Programming 

Filling polyhedral molds
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