Desagonale

Desagonale is a term used in geometry to describe a class of simple polygons that admit a non-crossing triangulation organized as a fan from a single vertex. Specifically, a polygon is desagonale if there exists a vertex such that all diagonals drawn from that vertex to all nonadjacent vertices lie entirely within the polygon and partition it into triangles. This implies the polygon is star-shaped with respect to that vertex and has a kernel that includes that vertex.

In standard language, the property is of interest in studies of polygon triangulations and visibility because it guarantees a straightforward, non-overlapping triangulation. All convex polygons are desagonale; some concave polygons are also desagonale if they have a kernel containing a vertex from which the entire boundary is visible.

Etymology: The form desagonale is formed from des- (Latin/Greek prefix) and gonal, drawing on the root for angle (gonia), mirroring terms like desagonal or desagonal shapes, though it is not widely used in formal literature.

Applications: Used in teaching triangulation algorithms, computer graphics, and computational geometry to illustrate fan triangulations and kernel concepts. It is sometimes used in problem sets or thought experiments to contrast with general polygon triangulations that do not have a consistent fan structure.

See also: polygon triangulation, kernel of a polygon, star-shaped polygon, fan triangulation.

Note: Desagonale is not a standard term in most mathematical texts; its use is primarily pedagogical or fictional in some sources.