Visibility polygon

In computational geometry, the visibility polygon or visibility region for a point p in the plane among obstacles is the possibly unbounded polygonal region of all points of the plane visible from p. The visibility polygon can also be defined for visibility from a segment, or a polygon. Visibility polygons are useful in robotics, video games, and in determining positions to locate facilities, such as the best placement of security guards in an art gallery.

If the visibility polygon is bounded then it is a star-shaped polygon. A visibility polygon is bounded if all rays shooting from the point eventually terminate in some obstacle. This is the case, e.g., if the obstacles are the edges of a simple polygon and p is inside the polygon. In the latter case the visibility polygon may be found in linear time.

Definitions

Formally, we can define the planar visibility polygon problem as such. Let be a set of obstacles (either segments, or polygons) in . Let be a point in that is not within an obstacle. Then, the point visibility polygon is the set of points in , such that for every point in , the segment does not intersect any obstacle in .