Duality in spherical trigonometry

I've written before about the law of cosines on a sphere:

cos(c) = cos(a) cos(b) + sin(a) sin(b) cos(C).

But that's just one of two laws of cosines on a sphere. The equation above is the law of cosinesfor sides. The law of cosines for angles is typographically very similar, just inverting the case of the letters used as arguments.

cos(C) = cos(A) cos(B) + sin(A) sin(B) cos(c).

Incidentally, the law of cosines for sides shows how the sides of a spherical triangle determine the angles. The law of cosines for angles shows how, unlike in the case of plane geometry, angles determine the sides.

Two kinds of angles

Let's back up and say what the letters refer to. As is conventional in plane geometry, the sides of a triangle area,b, andc, and the angles areA,B, andC, with a being the side opposite the angleA, and similarly for the other letters.

However, unlike plane geometry, the sides are also angles, but of a different sort. The sidesa,b, andcare central angles, and the angles"A,B, andC aredihedral angles.

The sides are arcs of a great circle. The sidea, for example, is a portion of a great circle, running between the vertices atB andC [1]. Think of a plane that contains the center of the sphere O and the verticesB andC. The length ofa is the angle atO.

The anglesA,B, andC are the angles at which great circles meet. In the formalism of differential geometry, these angles exist in the tangent plane at the vertices. These angles are intrinsic, measured along the (tangent plane of) the surface of the sphere. The central angles are extrinsic, measured in a plane that cuts through the sphere to its center.

Polar triangles

Now because sides are angles and vertices are angles, we can interchange the two kinds of angles to form thepolar triangle of the original triangle. This is a new triangle whose central angles are the complements of the original dihedral angles, and whose dihedral angles are the complements of the original central angles. In symbols,

a' = - A
b' = - B
c' = - C
A' = - a
B' = - b
C' = - c

Now we return to the laws of cosines at the top of the post: the law of cosines for angles is the law of cosines for sides applied to the polar triangle. Every equation in spherical trigonometry has a polar counterpart.

[1] There's some overloaded terminology here, just as in plane geometry. The symbol a is used to denote a side of a triangle and also the length of that side. The symbolA is used to denote both a vertex and the dihedral angle at that vertex.

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