Spherical Law of Cosines
Spherical Law of Cosines
Draw a spherical triangle on the surface of the unit sphere with center at the origin . Let the sides (arcs) opposite the vertices have lengths , and , and let be the angle at vertex . The spherical law of cosines is then given by , with two analogs obtained by permutations.
ABC
O
a
b
c
A
A
cosa=cosbcosc+sinbsinccosA
Details
Details
Let be the plane tangent to the sphere at , and let and . Then , , and . Express the length of in two ways using the usual planar law of cosines for the triangle in the plane :
p
A
Q=p⋂OB
R=p⋂OC
|OQ|=1/cosc
|AQ|=sinc/cosc
|OR|=1/cosb
|AR|=sinb/cosb
|QR|
AQR
p
|QR|=+-2cosA
2
sinb
cosb
2
sinc
cosc
sinb
cosb
sinc
cosc
With the triangle , the law of cosines gives
OQR
QR=+-2cosa
2
1
cosb
2
1
cosc
1
cosb
1
cosc
The two equalities give
cosa=cosbcosc+sinbsinccosA
References
References
[1] Wikipedia. "Spherical Law of Cosines." (Feb 22, 2017) en.wikipedia.org/wiki/Spherical_law_of _cosines.
External Links
External Links
Permanent Citation
Permanent Citation
Izidor Hafner
"Spherical Law of Cosines"
http://demonstrations.wolfram.com/SphericalLawOfCosines/
Wolfram Demonstrations Project
Published: February 23, 2017