Complexity of computing shortest paths in the plane with polygonal obstacles
Maybe I miss something, but if we consider the "easy" case, where all obstacles are points, then we have the problem of computing the shortest path between two vertices in a planar graph, which, if I am not wrong, it is known as sums-of-square-roots-hard.

PS. I wanted to add a comment and not an answer, but I can't find how. I apologize for that. Can the admins please help me with this.