Starshaped

Starshaped, or star-shaped, describes a property in geometry. A subset S of Euclidean space is star-shaped if there exists a point s0 in S such that for every s in S the line segment from s0 to s lies entirely within S. The point s0 is called a star center. A star-shaped set is “visible” from its star center, and the concept generalizes to shapes of any dimension.

In the plane, a polygon is star-shaped precisely when its kernel is nonempty. The kernel consists of all points from which the entire polygon is visible; it is the intersection of the inward-facing half-planes defined by the polygon’s edges. If the kernel is nonempty, the polygon is star-shaped; if it is empty, the polygon is not.

Relation to convexity: every convex polygon is star-shaped, since any point inside can serve as a star center. However, a non-convex polygon may be not star-shaped if its kernel is empty. The kernel itself is a convex region, even when the polygon is not.

Computation and extensions: Determining star-shapedness typically involves computing the kernel via half-plane intersection or clipping algorithms. In higher dimensions, the concept remains the same: a star-shaped set has at least one point from which every point of the set is reachable by a straight line contained in the set. The term is often used in computer graphics, robotics, and GIS to address visibility, illumination, and path-planning problems. Starshaped is commonly written with a hyphen, though starshaped is also encountered.