Angles as probabilities

Angles as pr obabilities David V . F eldman Daniel A. Klain Almost e veryone kno ws that the inner angles of a triang le sum to 180 o . But if you ask the typical mathematician ho w to sum the solid inner angles ove r the vertices of a tetrahedron, you are likely to recei ve a blank stare or a mystified shrug. In some cases you may be d irected to the Gram-Euler relations for hi gher dimensional polytopes [3, 4, 6, 7], a 19th century result unjustly consigned to relativ e obscurity . But th e answer is really much simpler than that, and here it is: The sum of the solid inner vertex angles of a tetrahedron T , divided by 2 π , giv es the probability that th e orthogonal projection of T onto a random 2-plane is a triangle. How simple is th at? W e will prove a m ore general theorem (Theorem 1) for simplices i n R n , but first consider th e analogous assertion in R 2 . The sum in radians of the angles of a triangle (2-simplex) T , when di vided by the length π of the unit semicircle, gives the probability that t he orthogon al projection of T on to a random li ne is a con ve x segment (1-simplex). Since t his is always th e case, th e probability is equal to 1, and the inner angle sum for ev ery triangle is the sam e. By contrast, a higher dimensional n -simplex m ay project one dimens ion d own either to an ( n − 1)-si mplex or to a lo wer dimensional con vex po lytope having n + 1 vertices. The inner angl e sum gives a measure of how often each of th ese possibili ties occurs. Let us make the notion o f “inner angle” more precise. Denote by S n − 1 the unit s phere i n R n centered at the origin. Recall that S n − 1 has ( n − 1)-dimensional volume (i.e. surface area) n ω n , where ω n = π n / 2 Γ (1 + n / 2) is the Euclidean volume of the unit ball in R n . Suppose that P is a con ve x polyto pe in R n , and let v be any poin t of P . The solid inner angle a P ( v ) of P at v is giv en by a P ( v ) = { u ∈ S n − 1 | v + ǫ u ∈ P for some ǫ > 0 } Let α P ( v ) denote the measure o f the solid angle a P ( v ) ⊆ S n − 1 , given by the usual surface area m easure o n s ubsets of the sphere. For the moment we are primarily concerned with v alues of α P ( v ) when v is a verte x of P . If u is a unit vector , then denote by P u the orthogonal pro jection of P onto the subspace u ⊥ in R n . Let v be a vertex of P . The projecti on v u lies in the relative interior of P u if and o nly if there exists ǫ ∈ ( − 1 , 1) such that v + ǫ u lies in the interior of P . This holds i ff u ∈ ± a P ( v ). If u is a random unit vector in S n − 1 , then (1) Probabi lity [ v u ∈ relativ e int erior( P u )] = 2 α P ( v ) n ω n , This giv es the probability that v u is no longer a verte x of P u . 1 2 For simplices we now obtain the follo wing theorem. Theor em 1 (Simplicial Angle S ums) . Let ∆ be an n-simple x in R n , and let u be a random un it vector . Denote by p ∆ the pr obability t hat the o rthogonal pr ojection ∆ u is an ( n − 1) -simplex. Then (2) p ∆ = 2 n ω n X v α ∆ ( v ) wher e the sum is taken ove r all vertices of the simplex ∆ . Pr oof. Since ∆ is an n -simpl ex, ∆ has n + 1 vertices, and a proj ection ∆ u has either n or n + 1 vertices. (Since ∆ u spans an a ffi ne space o f dim ension n − 1, it cannot hav e fe wer than n vertices.) In ot her words, either exactly 1 verte x of ∆ falls to the relati ve interior of ∆ u , so that ∆ u is an ( n − 1)-simplex, or none of them do. By the law of alternatives, the probabili ty p ∆ is now g iv en by t he sum of t he probabilities (1), taken ov er all v ertices of the simplex ∆ .  The probabili ty (2) is always equal to 1 for 2-dimensional si mplices (triangles). For 3-simplices (tetrahedra) the probabi lity may take any value 0 < p ∆ < 1. T o obtain a value closer to one, con sider the con vex h ull of an equ ilateral triangle i n R 3 with a point outside th e triang le, but very close to it s center . T o obtain p ∆ close to zero, consider the con ve x hull of two ske w line segments in R 3 whose centers are very close together (forming a tetrahedron t hat is almost a parallelog ram). Similarly , for n ≥ 3 t he so lid vertex angle sum of an n -simplex varies within a range 0 < X v α T ( v ) < n ω n 2 . Equality at either end is obtained o nly if one al lows for the degenerate l imiting cases. These bounds were obt ained earlier by Gaddum [1, 2], using a more com- plicated (and non-probabilisti c) approach. Similar considerations apply to the solid angles at arbitrary faces of con ve x polytopes. If F is a face of a con vex poly tope P , denot e the centroid of F by ˆ F , and d efine the solid inner angle measure α P ( F ) = α P ( ˆ F ), usin g the definitio n above for t he solid angle at a point in P . In analogy to (1), we hav e (3) Probabi lity [ ˆ F u ∈ relativ e int erior( P u )] = 2 α P ( F ) n ω n , Omitting cases of measures zero, this gi ves th e probability that a p roper face F is no longer a face of P u . (Note that dim F u = dim F for all directions u except a set of measure zero.) T aking complements, we ha ve (4) Probabi lity [ F u is a proper face of P u ] = 1 − 2 α P ( F ) n ω n . For 0 ≤ k ≤ n − 1, denote by f k ( P ) the number of k -dimension al fac es of a polytope P . The sum of the probabilities (4) gi ves the expected numb er of k -faces 3 of the projection of P onto a random hyperplane u ⊥ ; that is, (5) Exp [ f k ( P u )] = X dim F = k 1 − 2 α P ( F ) n ω n ! = f k ( P ) − 2 n ω n X dim F = k α P ( F ) , where the middle sum is taken ov er k -faces F of th e polytope P . If P is a con ve x polygon in R 2 , then P u is always a li ne segment with exactly 2 vertices, that is, f 0 ( P u ) = 2. In t his case the expectation identity (5) yields the familiar X v α P ( v ) = π ( f 0 ( P ) − 2) . If P is a con ve x polytope in R 3 , t hen P u is a con v ex polygo n, which alw ays has exactly as many vertices as edges; that is, f 0 ( P u ) = f 1 ( P u ). Therefore, Exp [ f 0 ( P u )] = Exp [ f 1 ( P u )], and the expectation identities (5) imply that 1 2 π X vertice s v α P ( v ) − 1 2 π X edge s e α P ( e ) = f 0 ( P ) − f 1 ( P ) = 2 − f 2 ( P ) . where the third equality follows from the class ical Euler formula f 0 − f 1 + f 2 = 2 for con v ex polyhedra in R 3 . These arguments were generalized b y Perles and Shephard [6] (see also [3, p. 315a]) to give a simpl e proof of the class ical Gram-Eul er identi ty for con ve x polytopes: (6) X F ⊆ ∂ P ( − 1) dim F α P ( F ) = ( − 1) n − 1 n ω n , where th e sum i s taken over all proper faces F of an n -dimensio nal con v ex poly- tope P . In the general case one applies the additivity of expectation to alternating sums o ver k of t he i dentities (5), obtaining ident ities t hat relate the Euler numbers of the boundaries ∂ P and ∂ P u . Since the ∂ P is an piece wise-linear ( n − 1)-sphere, while ∂ P u is a piecewise-linear ( n − 2)-sphere, th ese Euler nu mbers are easily computed, and (6) follows. The Gram-Euler i dentity (6) c an be vie wed as a discrete analog ue of th e Gauss- Bonnet theorem, a nd has been since generalized to Eu ler -type identities for angl e sums over polyto pes in spherical and hyperbolic spaces [3, 4, 7], as well as for mixed v olumes and other v aluations on polytopes [5]. R efere nces [1] J. W . Gadd um, The sums of the dihed ral and trihedral an gles in a tetrahedr on , Amer . Math. Monthly 59 (195 2), no. 6, 370–371. [2] , Distance sums on a spher e an d a ngle su ms in a simplex , Amer . M ath. Monthly 63 (1956 ), no. 2, 91–96. [3] B. Gr ¨ unbau m, Co n ve x Polytopes (2nd Ed.) , Springer V erlag, New Y ork, 2003. [4] P . McMullen, Non-linear angle-sum r ela tions for polyhedral con es and polytopes , Math. Proc. Camb . Phil. Soc. 78 (1975), 247–26 1. 4 [5] , V alua tions and Euler-type r elations on c ertain classes o f co n ve x polyto pes , Proc. Londo n M ath. Soc. 35 (1977 ), 113–135. [6] M. A. Perles an d G. C. Sh ephard , An gle sums of c onve x polyto pes , Math. Scand. 2 1 (1 967), 199–2 18. [7] D. M. Y . Sommerville, An Intr o duction to the Geometry of n Dime nsions , Dover Pub lications, New Y ork, 1983. Da vid V . Feldman: Dept. of Mathemati cs, Uni versity of New Hampshire , Dur ham, NH 03824 USA, David.Fe ldman@un h.edu Daniel A. Klain: Dept. o f Mat hematical Sciences, University of Massac husetts Lowell, Lowell, MA 01854 USA, Daniel Klain@u ml.edu