A probabilistic proof of the spherical excess formula
aa r X i v : . [ m a t h . HO ] S e p A probabilistic proof of the spherical excess formula
Daniel A. Klain
A triangle T in the unit sphere with inner angles θ , θ , and θ has area given by the spherical excess formula: Area ( T ) = θ + θ + θ − π. (1) θ θ θ F igure
1. A spherical triangle.This note o ff ers a probabilistic proof of the angle excess formula (1), based on theobservation that an unbounded cone at the origin in R has only three kinds of 2-dimensional orthogonal projections: a cone in R , a half-plane in R (this is an outcomeof probability zero), and all of R . See Figure 2. ? F igure
2. Projections of a 3-dimensional cone.Observe that, if we omit the middle outcome of measure zero, the number of edgeson each projected figure is twice the number of vertices. This formula was discovered in 1603 by Thomas Harriot [5, p. 65] and is also known as Girard’sformula [1, p. 95]. Some notation will help to interpret angles as probabilities. Let S denote the unitsphere in R centered at the origin, having surface area 4 π .Suppose that P is a convex polytope in R , and let x be any point of P . The solidinner angle a P ( x ) of P at x is given by a P ( x ) = { u ∈ S | x + ǫ u ∈ P for some ǫ > } . Let α P ( x ) denote the measure of the solid angle a P ( x ) ⊆ S , given by the usual surfacearea measure on subsets of the sphere.If F is a proper face of a convex polytope P , then the solid inner angle measure α P ( x )is the same at every point x in the relative interior of F . This value will be denoted by α P ( F ).Consider the case of an unbounded cone C with single vertex at the origin o , as inFigure 2. Specifically, let v , v , v be three linearly independent unit vectors in R , andlet C denote all non-negative linear combinations: C = { t v + t v + t v | t i ≥ } . The polyhedral cone C has exactly one vertex at o and three (unbounded) edges e i alongthe directions of the vectors v i . Note that α C ( o ) is the area of the spherical triangle withvertices at v i . Denote the inner angles of this triangle by θ i , as in Figure 1 (where o liesat the center of the sphere in Figure 1).Given a uniformly random unit vector u , let C u denote the orthogonal projection of C onto the plane u ⊥ . Evidently C u will resemble one of the outcomes in Figure 2.Specifically, C u will cover the entire plane u ⊥ i ff u lies in the interior of ± a C ( o ). Itfollows that C u = u ⊥ with probability Area ( a C ( o )) + Area ( − a C ( o ))4 π = α C ( o )4 π = α C ( o )2 π . Since the number of vertices of C u is either 0 or 1, the expected number of vertices of C u is given by the complementary probability E ( = − α C ( o )2 π . (2)Meanwhile, an edge e projects to the interior of C u i ff u lies in the interior of ± a C ( e ).Taking the complement as before, e projects to a boundary edge of C u with probability1 − α C ( e )2 π . Observe that each solid angle measure α C ( e i ) is given by 2 θ i (see Figure 3), sothat the expected number of edges of C u is E ( = X i − α C ( e i )2 π ! = X i (cid:18) − θ i π (cid:19) = − θ + θ + θ π . (3)Since the number of edges in C u is almost surely twice the number of vertices (seeFigure 2), the identities (2) and (3) imply that3 − θ + θ + θ π = E ( = E ( = − α C ( o ) π . (4) θ e F igure α C ( e ) = θ It is now immediate from (4) that α C ( o ) = θ + θ + θ − π, as asserted in (1). ❧ While the argument above generalizes easily to the case of a spherical polygon withmore than 3 sides, in higher dimensions there is a proliferation of cases that makes thisapproach much more complicated. However, a variation of this approach was appliedin [2] to an n -simplex ∆ to obtain the identity p ∆ = n ω n X v α ∆ ( v ) , where p ∆ is the probability that a random orthogonal projection ∆ u is an ( n − ω n denotes the volume of the unit ball in R n , and where the sum is taken over all vertices v of the n -simplex ∆ .Similar and more general arguments were used even earlier by Perles and Shephard[4] (see also [3, p. 315a] and [6]) to give a simple proof of the Gram-Euler identity forconvex polytopes, X F ⊆ ∂ P ( − dim F α P ( F ) = ( − n − n ω n , where the sum is taken over all proper faces F of an n -dimensional convex polytope P .R eferences [1] H. S. M. Coxeter. (1969). Introduction to Geometry , 2nd ed. New York: Wiley.[2] D. V. Feldman, D. Klain. (2009). Angles as probabilites.
Amer. Math. Monthly . 116(8):732–735.[3] B. Gr¨unbaum. (2003).
Convex Polytopes , 2nd ed. New York: Springer.[4] M. A. Perles, G. C. Shephard. (1967). Angle sums of convex polytopes.
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Geometry of Surfaces . New York: Springer.[6] E. Welzl. (1994). Gram’s equation — a probabilistic proof. In:
Results and Trends in TheoreticalComputer Science – Graz, 1994 , Lecture Notes in Comput. Sci., vol. 812. Berlin: Springer, pp. 422-424., Lecture Notes in Comput. Sci., vol. 812. Berlin: Springer, pp. 422-424.