Duality in spherical trigonometry

This evening I ran across an unexpected reference to spherical trigonometry: Thomas Hales’ lecture on lessons learned from the formal proof of the Kepler conjecture. He mentions at one point a lemma that was awkward to prove in its original form, but that became trivial when he looked at its spherical dual.

The sides of a spherical triangle are formed by great circular arcs through the vertices. Since the sides are portions of a circle, they can be measured as angles. So in spherical trig you have this interesting interplay of two kinds of angles: the angles formed at the intersections of the sides, and the angles describing the sides themselves.

Here’s how you form the dual of a spherical triangle. Suppose the vertices of the angle are AB, and C. Think of the arc connecting A and B as an equator, and let C‘ be the corresponding pole that lies on the same side of the arc as the original triangle ABC. Do the analogous process to find the points A‘ and B‘. The triangle ABC‘ is the dual of the triangle ABC. (This idea goes back to the Persian mathematician Abu Nasr Mansur circa 1000 AD.)

The sides in ABC‘ are the supplementary angles of the corresponding intersection angles in ABC, and the intersection angles in ABC‘ are the supplementary angles of the corresponding sides in ABC.

In his paper “Duality in the formulas of spherical trigonometry,” published in American Mathematical Monthly in 1909, W. A. Granville gives the following duality principle:

If the sides of a spherical triangle be denoted by Roman letters abc and the supplements of the corresponding opposite angles by the Greek letters α, β, γ, then from any given formula involving any of these six parts, we may wrote down a dual formula simply by interchanging the corresponding Greek and Roman letters.

Related: Notes on Spherical Trigonometry

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