Difference between revisions of "Triangle, defect of a"
From Encyclopedia of Mathematics
(Importing text file) |
(Added TeX done tag, references) |
||
| Line 1: | Line 1: | ||
| + | {{TEX|done}} | ||
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. | 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. | ||
| − | |||
| − | |||
| − | |||
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]]). | 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]]). | ||
| Line 11: | Line 9: | ||
====References==== | ====References==== | ||
| − | + | {| | |
| + | |- | ||
| + | |valign="top"|{{Ref|Be}}||valign="top"| M. Berger, "Geometry" , '''II''' , Springer (1987) pp. §19.5 | ||
| + | |- | ||
| + | |valign="top"|{{Ref|Co}}||valign="top"| H.S.M. Coxeter, "Introduction to geometry" , Wiley (1969) pp. §16.5 | ||
| + | |- | ||
| + | |} | ||
Revision as of 10:48, 22 April 2012
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.
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 |
How to Cite This Entry:
Triangle, defect of a. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Triangle,_defect_of_a&oldid=25037
Triangle, defect of a. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Triangle,_defect_of_a&oldid=25037
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article