angular defect

angular defect (plural angular defects)

1. (geometry, non-Euclidean geometry) 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.
   • 1864, Transactions of the Royal Society of Edinburgh, Volume 23, page 444,

     Prop. XVI. If the angular defects of two triangles are equal, the areas of the triangles are equal.

   • 1995, Arlan Ramsay, Robert D. Richtmyer, Introduction to Hyperbolic Geometry, page 120,

     We have seen that under any reasonable definition of area, the angular defect of a triangle is proportional to its area.

   • 1998, Kappa Mu Epsilon, The Pentagon: A Mathematics Magazine for Students‎, page 7:

     In hyperbolic geometry, similar triangles do not exist and the area of a triangle is directly proportional to its angular defect […] .

2. (geometry) The amount by which the total of the angles around a vertex of a polyhedron is less than 360° (2π radians).
   • 2005, Max K. Agoston, Computer Graphics and Geometric Modelling: Mathematics‎, page 604:

     The angular defect clearly has something to do with curvature, because the larger it is, the more pointed the surface is at the vertex.

   • 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°.

3. (dentistry, periodontics) The angular displacement of a tooth from vertical.
   • 1986, Fermin A. Carranza, Dorothy A. Perry, Clinical Periodontology for the Dental Hygienist, page 62,

     In most instances angular defects have accompanying infrabony pockets; infrabony pockets always have an underlying angular defect.

   • 2008, T. Siji Jacob, P. Arunmozhi, Viva Voce in Periodontics, page 74,

     The base of the angular defect is usually located apical to the surrounding bone and most often accompanied by infrabony pockets. […] Angular defects are classified on the basis of number of walls.

   • 2008, Dental Learning Systems, Compendium of Continuing Education in Dentistry, Volume 28, Issues 7-11, page 442,

     Without therapy, the positive predictive rate of an angular defect to forecast more bone loss (22 mm) during a 10-year study was 28%.

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.

• (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