Supplementary Solid Angles for Trihedron

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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.
Contributed by: Izidor Hafner (March 2017)
Open content licensed under CC BY-NC-SA
Snapshots
Details
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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