angular defect

Noun

 * 1)  The amount by which the total of the interior angles of a triangle is less than 180° (π radians); the amount by which the total of the internal angles of a polygon is less than what would be expected on the Euclidean plane.
 * 2)  The amount by which the total of the angles around a vertex of a polyhedron is less than 360° (2π radians).
 * 3) * 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.
 * 1) * 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°.
 * 1)  The angular displacement of a tooth from vertical.
 * 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.
 * 1) * 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°.
 * 1)  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

 * angular deficiency, angular deficit
 * angular deficiency, angular deficit
 * vertical defect