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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 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 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 on average 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 or over a triangle (with a minus) is precisely the curvature at that point or the total (integrated) over the triangle, as established by the Gauss–Bonnet theorem. == Defect of a vertex == For a polyhedron, the defect at a vertex equals 2π minus the sum of all the angles at the vertex (all the faces at the vertex are included). If a polyhedron is convex, then the defect of each vertex is always positive. If the sum of the angles exceeds a full turn, as occurs in some vertices of many non-convex polyhedra, then the defect is negative. The concept of defect extends to higher dimensions as the amount by which the sum of the dihedral angles of the cells at a peak falls short of a full circle. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Angular defect」の詳細全文を読む スポンサード リンク
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