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sin
R
sin
R
sin
R
sin A
ﾼ
sin B
ﾼ
sin C
:
The lengths of sides in the above equation are expressed by angles. The formula
can only be applied if the known lengths of sides are divided by the spherical radius
to get the angle at the center of the sphere. The calculated lengths of sides that are
expressed in degrees should be converted to lengths again because in practical cases
the sides are always represented by lengths. It often appears inconvenient to
convert. In the meantime, there also exist some round-off errors in computation
that will adversely influence the accuracy of computation.
We will attempt to find a simpler way to solve the triangle. Here, Legendre's
theorem is a simple and convenient way to solve the spherical triangle. The
Legendre method, in nature, is to solve the spherical triangle as the plane triangle
that has the same corresponding sides as the spherical triangle. The spherical angles
are required to make some simple changes.
As shown in Fig.
5.28
, let A
0
B
0
C
0
be a spherical triangle, a, b, and c are its three
sides, and
ε
00
denotes the spherical excess. We draw a planar triangle A
1
B
1
C
1
with
the same side lengths of a,b, and c. When the side lengths are not long, the three
internal angles in the two triangles can be proved to have the following relations:
9
=
00
A
1
ﾼ
A
0
ε
=
3
00
B
1
ﾼ
B
0
ε
=
3
,
ð
5
:
54
Þ
;
00
C
1
ﾼ
C
0
ε
=
3
00
00
,
ﾼ
R
2
ˁ
where
ε
ʔ
is the area of the plane triangle, and R denotes the radius of the
sphere.
The above equation is Legendre's theorem to solve spherical triangles. It shows
that one-third of the spherical excess of the given spherical triangle A
0
B
0
C
0
subtracted from each angle of the triangle gives the plane triangle A
1
B
1
C
1
whose
sides are equal in length to the corresponding sides of the spherical triangle. The
side lengths computed according to the formulae for the plane triangle will be the
side lengths of the spherical triangle.
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