This paper describes algorithms to compute Voronoi diagrams, shortest path maps, the Hausdorff distance, and the Fréchet distance in the plane with polygonal obstacles. The underlying distance measures for these algorithms are either shortest path distances or link distances. The link distance between a pair of points is the minimum number of edges needed to connect the two points with a polygonal path that avoids a set of obstacles. The motivation for minimizing the number of edges on a path comes from robotic motions and wireless communications because turns are more difficult in these settings than straight movements. Link-based Voronoi diagrams are different from traditional Voronoi diagrams because a query point in the interior of a Voronoi face can have multiple nearest sites. Our site-based Voronoi diagram ensures that all points in a face have the same set of nearest sites. Our distance-based Voronoi diagram ensures that all points in a face have the same distance to a nearest site. The shortest path maps in this paper support queries from any source point on a fixed line segment. This is a middle-ground approach because traditional shortest path maps typically support queries from either a fixed point or from all possible points in the plane. The Hausdorff distance and Fréchet distance are fundamental similarity metrics for shape matching. This paper shows how to compute new variations of these metrics using shortest paths or link-based paths that avoid polygonal obstacles in the plane.