Supplementary Solid Angles for Trihedron
Supplementary Solid Angles for Trihedron
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 , and are edges of a trihedron that determines the supplementary space angle.
OA
OB
OC
a
b
c
A
B
C
O'
A'
B'
C'
O'A'
O'B'
O'C'
The plane angles of the supplementary angle are , and , and its dihedral angles are , and .
π-A
π-B
π-C
π-a
π-b
π-c
The measure of the initial trihedral angle is (the spherical excess formula for a trihedron), while the measure of its supplementary angle is .
A+B+C-π
2π-(A+B+C)=(π-A)+(π-B)+(π-C)-π