Matches in DBpedia 2015-10 for { ?s ?p "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."@en }
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- Angular_defect abstract "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.".