Supplementary Solid Angles for Trihedron

point A

θ

1.047

point B

θ

1.047

ϕ

1.083

solid angle

edges of supplementary

solid angle

supplementary solid angle

additional labels

This Demonstration constructs a supplementary solid angle for a given trihedral solid angle. Let

OA

,

OB

and

OC

be the edges of a trihedron that determines the solid angle. The plane angles opposite the edges are denoted

a

,

b

,

c

and the dihedral angles at the edges are denoted

A

,

B

,

C

. Let

O'

be a point inside the trihedron and denote its orthogonal projections onto the faces of the trihedron by

A'

,

B'

and

C'

. Then

O'A'

,

O'B'

and

O'C'

are edges of a trihedron that determines the supplementary space angle.

The plane angles of the supplementary angle are

π-A

,

π-B

and

π-C

, and its dihedral angles are

π-a

,

π-b

and

π-c

.

The measure of the initial trihedral angle is

A+B+C-π

(the spherical excess formula for a trihedron), while the measure of its supplementary angle is

2π-(A+B+C)=(π-A)+(π-B)+(π-C)-π

.