Generalized Law of Sines -- from Wolfram MathWorld

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

gsinS=S-(KS^3)/(3!)+(K^2S^5)/(5!)-(K^3S^7)/(7!)+(K^4S^9)/(9!)-(K^5S^(11))/(11!)+....

(1)

For particular values of , we have

$gsinS={sum_(n=0)^(infty)((-1)^nS^(2n+1))/((2n+1)!) if K=1; sum_(n=0)^(infty)((-K)^nS^(2n+1))/((2n+1)!) if K>0; S+sum_(n=1)^(infty)0 if K=0; sum_(n=0)^(infty)((-K)^nS^(2n+1))/((2n+1)!) if K<0; sum_(n=0)^(infty)(S^(2n+1))/((2n+1)!) if K=-1,$

(2)

giving

gsinS={sinS if K=1; (sinSsqrt(K))/(sqrt(K)) if K>0; S if K=0; (sinhSsqrt(-K))/(sqrt(-K)) if K<0; sinhS if K=-1.

(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.