Interrelation between grid systems and star polygons of muqarnas ground projection plans

Recognizing S-Star Polygons

We consider the problem of recognizing star-polygons under staircase visibility (s-visibility). We show that the s-visibility polygon from a point inside a simple orthogonal polygon of n sides can be computed in O(n) time. When the polygon contains holes the algorithm runs in O(n log n) time, which we prove to be optimal by linear time reduction from sorting. We present an algorithm for computi...

K-star-shaped Polygons

We introduce k-star-shaped polygons, polygons for which there exists at least one point x such that for any point y of the polygon, segment xy crosses the polygon’s boundary at most k times. The set of all such points x is called the k-kernel of the polygon. We show that the maximum complexity (number of vertices) of the k-kernel of an n-vertex polygon is Θ(n) if k = 2 and Θ(n) if k ≥ 4. We giv...

Convexifying star-shaped polygons

Exploring Grid Polygons Online

We investigate the exploration problem of a short-sighted mobile robot moving in an unknown cellular room. To explore a cell, the robot must enter it. Once inside, the robot knows which of the 4 adjacent cells exist and which are boundary edges. The robot starts from a specified cell adjacent to the room’s outer wall; it visits each cell, and returns to the start. Our interest is in a short exp...

Exploring Simple Grid Polygons

We investigate the online exploration problem of a shortsighted mobile robot moving in an unknown cellular room without obstacles. The robot has a very limited sensor; it can determine only which of the four cells adjacent to its current position are free and which are blocked, i. e., unaccessible for the robot. Therefore, the robot must enter a cell in order to explore it. The robot has to vis...