In geometry, the (angular) defect (or deficit or deficiency) means the failure of some angles to add up to the expected amount of 360° or 180°, when such angles in the Euclidean plane would. The opposite notion is the excess.
Classically the defect arises in two ways:
the defect of a vertex of a polyhedron;
the defect of a hyperbolic triangle;
and the excess also arises in two ways:
the excess of a toroidal polyhedron.
the excess of a spherical triangle;
In the Euclidean plane, angles about a point add up to 360°, while interior angles in a triangle add up to 180° (equivalently, exterior angles add up to 360°). However, on a convex polyhedron the angles at a vertex add up to less than 360°, on a spherical triangle the interior angles always add up to more than 180° (the exterior angles add up to less than 360°), and the angles in a hyperbolic triangle always add up to less than 180° (the exterior angles add up to more than 360°).
In modern terms, the defect at a vertex is a discrete version of the curvature of the polyhedral surface concentrated at that point, and the Gauss–Bonnet theorem gives the total curvature as times the Euler characteristic , so the sum of the defects is .
In geometry, the (angular) defect (or deficit or deficiency) means the failure of some angles to add up to the expected amount of 360° or 180°, when such...
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excess), which is related to the sum of the dihedral angles, or by the angulardefect (or excess) of the vertex, which is related to the sum of the face angles...
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