angular defect

== English == === Noun === angular defect (plural angular defects) (geometry, non-Euclidean geometry) The amount by which the sum of the interior angles of a triangle is less than 180° (π radians); the amount by which the sum of the internal angles of a polygon is less than what would be expected on the Euclidean plane. (geometry) The amount by which the total of the angles around a vertex of a polyhedron is less than 360° (2π radians). 2014, C. R. Calladine, The Static-geometric Analogy in the Equations of Thin Shell Structures, W. Olszak, Thin Shell Theory: New Trends and Applications, page 294, Figure 4(c) shows a flattened view of a small part of the undeformed polygonalized S-surface, consisting of the triangles surrounding a particular vertex. When the S-surface is strained there will be a consequent change of angular defect, which we wish to calculate. 2015, Jan Guichelaar (translator and editor), Alex Van Den Brandhof, Arnout Jaspers (editors), Half a Century of Pythagoras Magazine, page 164, Theorem. For a spherical polyhedron the total angular defect equals 720°. (dentistry, periodontics) The angular displacement of a tooth from vertical. ==== Usage notes ==== The triangular/polygonal angular defect is zero on the Euclidean plane, but may, for example, be positive (meaning a deficit) in hyperbolic spaces or negative (an excess) in spherical geometry. In Euclidean space, the vertex angular defect is typically positive, but may be negative when the vertex is a saddle point, as may be the case on a toroidal polyhedron. ==== Synonyms ==== (amount by which the total of the interior angles of a polygon is less than on the Euclidean plane): angular deficiency, angular deficit (amount by which the total of the total of the angles around a vertex of a polyhedron is less than 360°): angular deficiency, angular deficit (angular displacement of tooth): vertical defect