Comparison triangles
Let
be the euclidean plane,
be the unit 2-sphere, and
be the hyperbolic plane. For
, let
and
denote the spaces obtained, respectively, from
and
by multiplying the distance by
. For any
,
is the unique complete, simply-connected, 2-dimensional Riemannian manifold of constant sectional curvature
.
Let
be a metric space. Let
be a geodesic triangle in
, i.e. three points
,
and
and three geodesic segments
,
and
. A comparison triangle
in
for
is a geodesic triangle in
with vertices
,
and
such that
,
and
.
Such a triangle, when it exists, is unique up to isometry. The existence is always true for
. For
, it can be ensured by the additional condition
(i.e. the length of the triangle does not exceed that of a great circle of the sphere
).
Comparison angles
The interior angle of
at
is called the comparison angle between
and
at
. This is well-defined provided
and
are both distinct from
, and only depends on the lengths
. Let it be denoted by
. Using inverse trigonometry, one has the formulas:


Alexandrov angles
Comparison angles provide notions of angles between geodesics that make sense in arbitrary metric spaces. The Alexandrov angle, or outer angle, between two nontrivial geodesics
with
is defined as
Comparison tripods
The following similar construction, which appears in certain possible definitions of Gromov-hyperbolicity, may be regarded as a limit case when
.
For three points
in a metric space
, the Gromov product of
and
at
is half of the triangle inequality defect:
Given a geodesic triangle
in
with vertices
, the comparison tripod
for
is the metric graph obtained by gluing three segments
of respective lengths
along a vertex
, setting
.
One has
and
is the union of the three unique geodesic segments
. Furthermore, there is a well-defined comparison map
with
such that
is isometric on each side of
. The vertex
is called the center of
, and its preimage under
is called the center of
, its points the internal points of
, and its diameter the insize of
.
One way to formulate Gromov-hyperbolicity is to require
not to change the distances by more than a constant
. Another way is to require the insizes of triangles
to be bounded above by a uniform constant
.
Equivalently, a tripod is a comparison triangle in a universal real tree of valence
. Such trees appear as ultralimits of the
as
.[1]