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Difference between revisions of "Triangle, defect of a"

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The amount by which the sum of the internal angles of a triangle on the Lobachevskii plane is short of two right angles. The defect of a triangle is proportional to its area. The maximum value of the defect of a triangle equals two right angles if all the vertices of the triangle are points at infinity.
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The corresponding notion for geodesic triangles on general surfaces of negative curvature is contained in the Gauss–Bonnet integral formula (cf. [[Gauss–Bonnet theorem|Gauss–Bonnet theorem]]).
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The amount by which the sum of the internal angles of a triangle on the Lobachevskii plane is short of two right angles: cf. [[Lobachevskii geometry]]. The defect of a triangle is proportional to its area. The maximum value of the defect of a triangle equals two right angles if all the vertices of the triangle are points at infinity.
  
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For a triangle in [[Spherical geometry|spherical geometry]] one has the opposite effect that the sum of the angles is greater than two right angles, the spherical excess or angular excess.
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The corresponding notion for geodesic triangles on general surfaces of negative curvature is contained in the Gauss–Bonnet integral formula (cf. [[Gauss–Bonnet theorem]]).
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For a triangle in [[spherical geometry]] one has the opposite effect that the sum of the angles is greater than two right angles, the spherical excess or angular excess.
  
 
The defect of a triangle is also called angular defect or hyperbolic defect.
 
The defect of a triangle is also called angular defect or hyperbolic defect.
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|valign="top"|{{Ref|Be}}||valign="top"|  M. Berger,  "Geometry" , '''II''' , Springer  (1987)  pp. §19.5
 
|valign="top"|{{Ref|Be}}||valign="top"|  M. Berger,  "Geometry" , '''II''' , Springer  (1987)  pp. §19.5
 
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|valign="top"|{{Ref|Co}}||valign="top"|  H.S.M. Coxeter,  "Introduction to geometry" , Wiley  (1969)  pp. §16.5
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|valign="top"|{{Ref|Co}}||valign="top"|  H.S.M. Coxeter,  "Introduction to geometry" , Wiley  (1969)  pp. §16.5 {{ZBL|0181.48101}}
 
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Latest revision as of 11:52, 13 December 2015

2020 Mathematics Subject Classification: Primary: 51M10 [MSN][ZBL]

The amount by which the sum of the internal angles of a triangle on the Lobachevskii plane is short of two right angles: cf. Lobachevskii geometry. The defect of a triangle is proportional to its area. The maximum value of the defect of a triangle equals two right angles if all the vertices of the triangle are points at infinity.

The corresponding notion for geodesic triangles on general surfaces of negative curvature is contained in the Gauss–Bonnet integral formula (cf. Gauss–Bonnet theorem).

For a triangle in spherical geometry one has the opposite effect that the sum of the angles is greater than two right angles, the spherical excess or angular excess.

The defect of a triangle is also called angular defect or hyperbolic defect.

References

[Be] M. Berger, "Geometry" , II , Springer (1987) pp. §19.5
[Co] H.S.M. Coxeter, "Introduction to geometry" , Wiley (1969) pp. §16.5 Zbl 0181.48101
How to Cite This Entry:
Triangle, defect of a. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Triangle,_defect_of_a&oldid=36909
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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