angular defect

angular defect (plural angular defects)

1. (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.
   • 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, numbers 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