Define
M
k
2
as the 2-dimensional metric space of constant curvature
k
. So, for example,
M
0
2
is the Euclidean plane,
M
1
2
is the surface of the unit sphere, and
M
−
1
2
is the hyperbolic plane.
Let
X
be a metric space. Let
T
be a triangle in
X
, with vertices
p
,
q
and
r
. A comparison triangle
T
∗
in
M
k
2
for
T
is a triangle in
M
k
2
with vertices
p
′
,
q
′
and
r
′
such that
d
(
p
,
q
)
=
d
(
p
′
,
q
′
)
,
d
(
p
,
r
)
=
d
(
p
′
,
r
′
)
and
d
(
r
,
q
)
=
d
(
r
′
,
q
′
)
.
Such a triangle is unique up to isometry.
The interior angle of
T
∗
at
p
′
is called the comparison angle between
q
and
r
at
p
. This is well-defined provided
q
and
r
are both distinct from
p
.