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Supplementary Solid Angles for Trihedron

point A

1.047

point B

1.047

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

and

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

and the dihedral angles at the edges are denoted

. Let

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

and

. 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

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