On the separation of a polyhedron from its single-part mold

Casting is a manufacturing process where liquid material is poured into a mold having the shape of a desired product. After the material solidifies, the product is pulled out of the mold. We study the case in which the mold is made of a single part and the object to be produced is a three-dimensional polyhedron. Objects that can be produced this way are called castable with a single-part mold. A direction in which the object can be removed without breaking the mold is called a valid pull-out direction. We give an algorithm that decides whether a given polyhedron with n facets is castable with a single-part mold, and if so indicates how to orient the polyhedron in the mold and a direction in which the product can be pulled out without breaking the mold. Our algorithm runs in O(n) time. The best previously known algorithm for this problem runs in O(n
²
) time.