Computing the constrained Euclidean, geodesic and link centre of a simple polygon with applications

In the manufacturing industry, finding a suitable location for the pin gate (the pin gate is the point from which liquid is poured or injected into a mold) is a difficult problem when viewed from the fluid dynamics of the molding process. However, experience has shown that a suitable pin gate location possesses several geometric characteristics, namely the distance from the pin gate to any point in the mold should be small and the number of turns on the path from a point in the mold to the pin gate should be small. We address the problem of computing locations that possess these geometric characteristics. Given a mold M (modeled by an n vertex simple polygon) we show how to compute the Euclidean center of M constrained to lie in the interior of M or on the boundary of M in O (n log n + k) time where k is the number of intersections between M and the furthest point Voronoi diagram of the vertices of M. We show how to compute the geodesic center of M constrained to the boundary in O (n log n) time and the geodesic center of M constrained to lie in a polygonal region in O (n(n + k)) time. Finally, we show how to compute the link center of M constrained to the boundary of M in O (n log n) time.