The generalized law of sines applies to a simplex in space of any dimension with constant Gaussian curvature. Let us work up
to that. Initially in two-dimensional space, we define a generalized sine function
for a one-dimensional simplex (line segment) with content (length) 
in space of constant Gaussian
curvature 
as
 |
(1)
|
For particular values of 
, we have
 |
(2)
|
giving
 |
(3)
|
Thus in elliptic space (
), the function is the sine function; in Euclidean space
(
),
the function is simply the content itself; and in hyperbolic space (
), the function is the hyperbolic sine function. Thus for
a two-dimensional simplex 
with edges of length 
, 
, and 
, we can express the law of sines for space with any constant
Gaussian curvature as
 |
(4)
|
For Euclidean space (
), equation (4) specializes to
 |
(5)
|
For the elliptic plane or the unit sphere (
), equation (4) specializes to
 |
(6)
|
For the hyperbolic plane (
), equation (4) specializes to
 |
(7)
|
Our generalization for the two-dimensional law of sines is not complete, however, because we have not defined what the ratio is equal to, and that requires that we define a generalized sine function for a two-dimensional simplex.
Suppose that 
is a two-dimensional simplex (triangle) in space of constant Gaussian curvature 
,
and that we have defined the generalized sine function for such a simplex, 
. Let the vertices of 
be labeled 
and the opposing edges 
. Then the generalized law of sines is expressed as
 |
(8)
|
This equation can be used to calculate 
; its value is
 |
(9)
|
where the denominator on the right will cancel with one of the factors of the numerator.
For an 
-dimensional
simplex 
in space of constant Gaussian curvature 
with vertex 
opposite facet 
, the law of sines may be expressed as
 |
(10)
|
where 
is the 
-dimensional
sine of the vertex angle of the simplex at vertex 
, and 
is defined as
 |
(11)
|
For a right simplex, the sine of the right angle is one, so that the sine of any vertex angle of a right simplex is the ratio of the generalized sine function of the opposite facet to the generalized sine function of the facet opposite the right angle.
In elliptic space (
), the generalized sine function is the polar sine of the
simplex. In Euclidean space (
), the function is 
times the content of an 
-dimensional simplex. In hyperbolic space (
), the function is the hyperbolic polar sine of the simplex.
Thus, we can specialize equation (10) for elliptic space (
)
to
 |
(12)
|
We can specialize equation (10) for Euclidean space (
) to
 |
(13)
|
Finally, we can specialize equation (10) for hyperbolic space (
)
to
 |
(14)
|
