11246

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.
The plane angles of the supplementary angle are , and , and its dihedral angles are , and .
The measure of the initial trihedral angle is (the spherical excess formula for a trihedron), while the measure of its supplementary angle is .

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This Demonstration gives an animation for Figure 5.5 in [3, p. 186].
The deficiency of a solid angle determined by an -sided spherical polygon with angles , , …, is . The deficiency of a solid angle equals its supplementary angle [3, pp. 186–187].
References
[1] Wikipedia. "Spherical Law of Cosines." (Feb 23, 2017) en.wikipedia.org/wiki/Spherical_law_of _cosines.
[2] Wikipedia. "Spherical Trigonometry." (Feb 23, 2017) en.wikipedia.org/wiki/Spherical_trigonometry.
[3] P. R. Cromwell, Polyhedra, New York: Cambridge University Press, 1997.
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