Comparison triangle
Encyclopedia
Define 
as the 2-dimensional metric space
of constant curvature
. So, for example, 
is the Euclidean plane, 
is the surface of the unit sphere
, and
is the hyperbolic plane
.
Let
be a metric space
. Let
be a triangle
in
, with vertices 
, 
and 
. A comparison triangle 
in 
for 
is a triangle in 
with vertices 
, 
and 
such that 
, 
and 
.
Such a triangle is unique up to isometry
.
The interior angle of
at 
is called the comparison angle between 
and 
at 
. This is well-defined provided 
and 
are both distinct from 
.
as the 2-dimensional metric spaceMetric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
of constant curvature
Constant curvature
In mathematics, constant curvature in differential geometry is a concept most commonly applied to surfaces. For those the scalar curvature is a single number determining the local geometry, and its constancy has the obvious meaning that it is the same at all points...
. So, for example, is the Euclidean plane, is the surface of the unit sphereUnit sphere
In mathematics, a unit sphere is the set of points of distance 1 from a fixed central point, where a generalized concept of distance may be used; a closed unit ball is the set of points of distance less than or equal to 1 from a fixed central point...
, and
is the hyperbolic planeHyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...
.
Let
be a metric spaceMetric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
. Let
be a triangleTriangle
A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....
in
, with vertices , and . A comparison triangle in for is a triangle in with vertices , and such that , and .Such a triangle is unique up to isometry
Isometry
In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...
.
The interior angle of
at is called the comparison angle between and at . This is well-defined provided and are both distinct from .