Mapping Polygons

This thesis is concerned with simple agents that move from vertex to vertex along straight lines inside a simple polygon with the goal of reconstructing the visibility graph. The visibility graph has a node for each vertex of the polygon with an edge between two nodes if the corresponding vertices see each other, i.e., if they can be connected by a straight line inside the polygon. While at a vertex, the agent perceives all vertices visible to its current location in the order in which they appear along the boundary. In each step, the agent can choose one of these vertices and move there. We show that an agent that can distinguish whether two visible vertices are neighbors on the boundary cannot always solve the visibility graph reconstruction problem when restricted to moving along the boundary only. This even remains true if the agent knows the total number of vertices beforehand and if it can measure the angles formed by the boundary of the polygon. On the other hand, we show that an agent that can measure the angles between edges of the visibility graph can always solve the visibility graph reconstruction problem, even when restricted to moving along the boundary only. We further consider an angle-type sensor which allows to distinguish whether the angle between any two edges is convex or reflex, and a look-back sensor which allows the agent to move back to where it came from in its last move. We show that an agent equipped with both sensors can always solve the visibility graph reconstruction problem, even without prior knowledge about the total number of vertices. The same is true for an agent that can measure the angle between any two edges and has a compass. For agents that have knowledge of an upper bound on the total number of vertices, we show stronger results. We show that in this setting an agent with look-back sensor or angle-type sensor can always solve the visibility graph reconstruction problem. We further show that multiple, identical, deterministic, indistinguishable such agents can find each other in any polygon.