Rotational castability of simple polygons

Stefanie Wuhrer

Abstract

A popular manufacturing technique is to pour liquid into a cast and to remove the cast once the liquid has hardened. We consider the case where the cast is modeled by a simple polygon and where the cast consists of exactly two parts. Removing the two cast parts using translations has been studied extensively. In this talk, the problem of rotational castability is discussed. That is, the following two problems are addressed:

1. Given a center of rotation r in the plane, we determine whether there exists a partitioning of the cast into exactly two parts, such that one part can be rotated clockwise around r and the other part can be rotated counterclockwise around r without colliding with the interior of the polygon. We present an algorithm to solve this problem with running time that is linear in the number of vertices of the polygon.
2. An algorithm is presented to find all the points in the plane that allow a cast partitioning as described above. The algorithm’s running time is quadratic in the number of vertices of the polygon.