Relations between Plane Angles and Solid Angles in a Trihedron
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Let ,
and
be the edges of a trihedron that determines a solid angle. The plane angles opposite the edges are denoted
,
,
, and the angles between the edges and their opposite faces are denoted
,
,
. Construct three planes parallel to the faces
,
and
at distance 1 from the corresponding faces. Let the intercepts of these planes with edges of the solid angle be
,
,
. Also define the points
,
,
,
such that
,
,
,
, to get a parallelepiped with all faces of equal area, since all heights are equal. The lengths of the edges are
,
and
. The areas of the faces are
,
and
.
Contributed by: Izidor Hafner (February 2017)
Open content licensed under CC BY-NC-SA
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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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