 We define certain properties such a pair of polyhedron and its snapping fixture may possess, and prove two theorems related to the pair. We introduce an algorithm that produces a snapping fixture with such properties in O(n ^5) time , if a snapping fixture exists , and an efficient and robust implementation of this algorithm .
<meaning-changed>  We define certain properties such a pair of polyhedron and its snapping fixture may possess, and prove two theorems related to the pair. We introduce an algorithm that produces a snapping fixture with such properties in O(n ^5) time , if a snapping fixture exists , and an efficient and robust implementation of this algorithm .
 We prove the minimal closure conditions under which such fixtures can hold parts, using Helly's theorem. We then introduce an algorithm that produces a snapping fixture with such properties in O(n ^5) time , if a snapping fixture exists , and an efficient and robust implementation of this algorithm .