angulardefect (pluralangular defects)
- (geometry, non-Euclidean geometry) The amount by which the sum of the interior angles of atriangle is less than 180° (πradians); the amount by which the sum of the internal angles of apolygon is less than what would be expected on theEuclidean plane.
1864,Transactions of the Royal Society of Edinburgh, volume23,page444:Prop. XVI.If theangular defects of two triangles are equal, the areas of the triangles are equal.
1995, Arlan Ramsay, Robert D. Richtmyer,Introduction to Hyperbolic Geometry,page120:We have seen that under any reasonable definition of area, theangular defect of a triangle is proportional to its area.
1998, Kappa Mu Epsilon,The Pentagon: A Mathematics Magazine for Students[1], page 7:In hyperbolic geometry, similar triangles do not exist and the area of a triangle is directly proportional to itsangular defect[…].
- (geometry) The amount by which the total of the angles around avertex of apolyhedron is less than 360° (2π radians).
2005, Max K. Agoston,Computer Graphics and Geometric Modelling: Mathematics[2], page604:Theangular 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 polygonalizedS-surface, consisting of the triangles surrounding a particular vertex. When theS-surface is strained there will be a consequent change ofangular 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 totalangular defect equals 720°.
- (dentistry, periodontics) The angular displacement of a tooth from vertical.
1986, Fermin A. Carranza, Dorothy A. Perry,Clinical Periodontology for the Dental Hygienist,page62:In most instancesangular defects have accompanying infrabony pockets; infrabony pockets always have an underlyingangular defect.
2008, T. Siji Jacob, P. Arunmozhi,Viva Voce in Periodontics,page74:The base of theangular 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, volume28, numbers7-11,page442:Without therapy, the positive predictive rate of anangular defect to forecast more bone loss (22 mm) during a 10-year study was 28%.
Thetriangular/polygonalangular defect is zero on theEuclidean plane, but may, for example, be positive (meaning a deficit) inhyperbolic spaces or negative (an excess) inspherical geometry.
InEuclidean space, thevertexangular defect is typically positive, but may be negative when the vertex is asaddle point, as may be the case on atoroidal polyhedron.
