Relations between Plane Angles and Solid Angles in a Trihedron
Relations between Plane Angles and Solid Angles in a Trihedron
Let , and be the edges of a trihedron that determines a solid angle. The plane angles opposite the edges are denoted , , , and the angles between the edges and their opposite faces are denoted , , . Construct three planes parallel to the faces , and at distance 1 from the corresponding faces. Let the intercepts of these planes with edges of the solid angle be , , . Also define the points , , , such that =+, =+, =+, =++, to get a parallelepiped with all faces of equal area, since all heights are equal. The lengths of the edges are , and . The areas of the faces are , and .
OA
OB
OC
α
β
γ
a
b
c
OBC
OAC
OAB
A'
B'
C'
O'
P
Q
R
OR
OA'
OC'
OP
OB'
OC'
OQ
OB'
OA'
OO'
OA'
OB'
OC'
|OA'|=1/sina
|OB'|=1/sinb
|OC'|=1/sinc
sinα/sinbsinc
sinβ/sinasinc
sinγ/sinbsina
Since the areas are equal, .
sinasinα=sinbsinβ=sincsinγ