Editing Angular defect (section)

In [[geometry]], the '''angular defect''' or simply '''defect''' (also called '''deficit''' or '''deficiency''') is the failure of some [[angle]]s to add up to the expected amount of 360° or 180°, when such angles in the [[Euclidean plane]] would. The opposite notion is the [[angle excess|''excess'']].

Classically the defect arises in two contexts: in the Euclidean plane, angles about a point add up to 360°, while [[Internal and external angle|interior angles]] in a triangle add up to 180°. However, on a [[Polyhedron|convex polyhedron]], the angles of the faces meeting at a vertex add up to ''less'' than 360° (a defect), while the angles at some vertices of a [[nonconvex polyhedron]] may add up to ''more'' than 360° (an excess). Also the angles in a [[hyperbolic triangle]] add up to ''less'' than 180° (a defect), while those on a [[spherical triangle]] add up to ''more'' than 180° (an excess).

In modern terms, the defect at a vertex is a discrete version of the [[Gaussian curvature|curvature]] of the polyhedral surface [[Dirac delta function|concentrated at that point]]. Negative defect indicates that the vertex resembles a [[saddle point]] (negative curvature), whereas positive defect indicates that the vertex resembles a [[local maximum]] or minimum (positive curvature). The [[Gauss–Bonnet theorem]] gives the total curvature as <math>2\pi</math> times the [[Euler characteristic]] <math>\chi = 2</math>, so for a convex polyhedron the sum of the defects is <math>4\pi</math>, while a [[toroidal polyhedron]] has <math>\chi = 0</math> and total defect zero.