Sumangle

Sumangle, short for sum angle, is a concept used in geometry to denote the total angular change experienced when traversing a polygonal chain. It is particularly relevant in planar geometry and computational geometry for describing how a path turns as it proceeds along consecutive edges.

Definition. Given a polygonal chain with vertices v0, v1, ..., v_{n-1}, the exterior turning angle at vertex i is the directed angle from edge (v_i, v_{i+1}) to edge (v_{i+1}, v_{i+2}) measured with a consistent orientation. The sumangle is the sum over i of these turning angles, commonly treated as a signed quantity. If the chain is closed and simple, the sumangle equals ±2π, with the sign determined by the orientation (positive for counterclockwise, negative for clockwise).

Special cases and relations. For a simple, non-self-intersecting polygon with n vertices, the sum of exterior turning angles is 2π, while the sum of interior angles is (n−2)π. For self-intersecting polygons, the sumangle can reflect the winding behavior of the path and is typically related to the polygon’s turning number or winding number around a point. The concept is related to total curvature in discrete geometry.

Computational use. Sumangle is used in algorithms that analyze polygon orientation, robustness of geometric predicates, and curvature discretization on polygonal meshes. It can be computed in tandem with other polygon properties and is sometimes preferred in discussions of angular changes over interior angle sums alone.

Notes. The term sumangle is not universally standardized; in many sources the same idea is described using turning angles or total curvature rather than under this fixed label.