starconvexity

Starconvexity is a property of certain geometric shapes. A set is starconvex if there exists at least one point within the set from which all other points in the set are visible. Visibility here means that the line segment connecting the two points lies entirely within the set. The set of points from which this visibility is possible is called the star-center. A star-convex set is a generalization of a convex set. In a convex set, every point can see every other point, meaning any point within the set can serve as a star-center. However, a star-convex set may have only a limited region or even a single point as its star-center.

The concept of starconvexity is important in various fields, including computer graphics, image processing, and computational geometry. For instance, in computer graphics, it can be used to define regions for rendering or collision detection. In image processing, it might be relevant for analyzing object shapes. The star-center property allows for certain algorithms that rely on a central point of reference within a non-convex but star-shaped region.

Examples of star-convex sets include a simple convex polygon, a star shape formed by connecting vertices of a regular polygon to its center, or even a simple circle. An annulus, which is the region between two concentric circles, is not star-convex because there are points in the annulus that cannot see all other points, as the line segment would pass through the hole in the center. The existence of a star-center guarantees a degree of "connectedness" or accessibility within the set, even if the set itself is not fully convex.