Comparison triangle explained
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
.
References
