This is the accepted version of the journal article: Cufí Sobregrau, Julià,; Gallego Gómez, Eduardo; Reventós, Agustí. «Integral geometry about the visual angle of a convex set». Rendiconti del Circolo Matem- atico di Palermo, Vol. 69 (October 2019), p. 1115–1130. DOI 10.1007/s12215- 019-00461-w This version is available at https://ddd.uab.cat/record/288241 under the terms of the license Noname manuscript No. (will be inserted by the editor) Integral Geometry about the visual angle of a convex set Julià Cufı́ · Eduardo Gallego · Agustı́ Reventós Received: / Accepted: Abstract In this paper we deal with a general type of integral formulas of the visual angle, among them those of Crofton, Hurwitz and Masotti, from the point of view of Integral Geometry. The purpose is twofold: to provide an interpretation of these formulas in terms of integrals of functions with respect to the canonical density in the space of pairs of lines and to give new simpler proofs of them. Keywords Convex set · Visual angle · Densities Mathematics Subject Classification (2010) 52A10, 53A04 1 Introduction Throughout this paper K will be a compact convex set in R2 with boundary a curve of class C 1 . We will denote by F the area of K and by L the length of its boundary. In 1868 Crofton showed ([1]), using arguments that nowadays belong to Integral Geometry, the well known formula Z 2 (ω − sin ω )dP + 2πF = L2 , (1) P ∈K / where ω = ω (P ) is the visual angle of K from the point P , that is the angle between the two tangents from P to the boundary of K . In terms of Integral Geometry both sides of this formula represent the measure of pairs of lines meeting K . In fact the normalized measure of all pairs of lines meeting K is L2 , twice the integral of ω − sin ω with respect to the area element dP is the measure of those pairs of lines The authors were partially supported by grants 2017SGR358, 2017SGR1725 (Generalitat de Catalunya) and PGC2018-095998-B-100 (Ministerio de Economı́a y Competitividad) Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia. E-mail: jcufi@mat.uab.cat, egallego@mat.uab.cat, agusti@mat.uab.cat 2 Julià Cufı́ et al. intersecting themselves outside K and 2πF is the measure of those intersecting themselves in K . Later on, Hurwitz in 1902, in his celebrated paper [4] on the application of Fourier series to geometric problems, considers the integral of some new functions of the visual angle. Specifically he proves Z fk (ω )dP = L2 + (−1)k π 2 (k2 − 1)c2k , (2) P ∈K / where k+1 k−1 fk (ω ) = −2 sin ω + sin((k − 1)ω ) − sin((k + 1)ω ), k ≥ 2, (3) k−1 k+1 and c2k = a2k + b2k , with ak , bk the Fourier coefficients of the support function of K . In the particular case k = 2 formula (2) gives Z 3 9 sin3 ω dP = L2 + π 2 c22 . (4) P ∈K / 4 4 Masotti Biggiogero in 1955 ([5]) states without proof the following Crofton’s type formula 4L2 X 1  2 Z (ω 2 − sin2 ω ) dP = −π 2 F + + 8π c . (5) P ∈K / π 1 − 4k2 2k k≥1 In [2] a unified approach that encompasses the previous results is provided. As well the following formula for the integral of any power of the sine function of the visual angle, that generalizes (4), is given: Z m! sinm (ω ) dP = L2 P ∈K / 2m (m − 2)Γ ( m− 2 1 2 ) k m!π 2 X (−1) 2 +1 (k2 − 1) + m+1−k k c2 . (6) 2m−1 (m − 2) Γ ( m+1+k 2 ) Γ ( 2 ) k≥2,even In this paper we deal with a general type of integral formulas of the visual angle including those we have just commented above, from the point of view of Integral Geometry according to Crofton and Santaló [6]. The purpose is twofold: to provide an interpretation of these formulas in terms of integrals of functions with respect to the canonical density in the space of pairs of lines and to give new simpler proofs of them. For each straight line G of the plane that does not pass through the origin let P be the point of G at a minimum distance from the origin. We take as coordinates for G the polar coordinates (p, ϕ) of the point P , with p > 0 and 0 ≤ ϕ < 2π . Notice that p and ϕ can also be seen as functions in this space of lines, and we shall write p(G), ϕ(G) for the corresponding coordinates of the straight line G. The invariant density in the set of lines of the plane not containing the origin is given by a constant multiple of dG = dp ∧ dϕ. In fact this density is, except for a constant factor, the only one invariant under Euclidean motions (see [6], Section Integral Geometry about the visual angle of a convex set 3 I.3.1). In the space of ordered pairs of lines we consider the canonical density dG1 ∧ dG2 . For every function f˜(G1 , G2 ) integrable with respect to dG1 ∧ dG2 we can consider the density f˜(G1 , G2 ) dG1 ∧ dG2 . We prove in Proposition 1 that this density is invariant under Euclidean motions if and only if f˜(G1 , G2 ) = f (ϕ(G2 ) − ϕ(G1 )) with f an even π -periodic function on R. For such densities it follows from Theorem 1 and Corollary 2 that X Z A0 L2 + π 2 c22n A2n = f (ϕ(G2 ) − ϕ(G1 )) dG1 ∧ dG2 = n≥1 Gi ∩K6=∅ Z = 2 H (π )F + 2 H (ω ) dP, (7) P ∈K / where Ak , k ≥ 0, are the Fourier coefficients of f corresponding to cos(kϕ), and H is the C 2 function on [0, π ] satisfying H 00 (x) = f (x) sin(x), x ∈ [0, π ], and H (0) = H 0 (0) = 0. The above two equalities are the main tools to obtain both new proofs of the previous formulas and their interpretation as integrals of functions with respect to the canonical density in the space of pairs of lines. As concerning to this second point, in section 3.3 one obtains the following formulas. - Crofton’s formula Z Z 1 (ω − sin ω ) dP = −πF + dG1 ∧ dG2 P ∈K / 2 Gi ∩K6=∅ - Hurwitz’s formula Z Z fk (ω ) dP = (1 + (−1)k (k2 − 1) cos(k(ϕ(G2 ) − ϕ(G1 )))) dG1 ∧ dG2 P ∈K / Gi ∩K6=∅ - Masotti’s formula Z Z (ω 2 − sin2 ω ) dP = −π 2 F + 2 | sin(ϕ(G2 ) − ϕ(G1 ))|dG1 ∧ dG2 P ∈K / Gi ∩K6=∅ - Power sine formula Z sinm ω dP = P ∈K / Z 1   m(m − 1)| sinm−3 (ϕ(G2 ) − ϕ(G1 ))| − m2 | sinm−1 (ϕ(G2 ) − ϕ(G1 ))| dG1 ∧dG2 2 Gi ∩K6=∅ Moreover using the first equality in (7) one gets the announced new proofs of formulas (1), (5) and (6). Concerning Hurwitz’s integral, when we apply the methods here developed, there appears a different behavior according to whether k is even or odd. For k even using (7) one gets a new proof of (2). Nevertheless when k is odd the function associated to the Hurwitz integral is not π -periodic since the function cos(kx) is not, and so we cannot use (7). In this case appealing to Proposition 5 one obtains a new result that involves a decomposition of the visual angle ω into two parts ω = ω1 + ω2 that also have a geometrical interpretation. 4 Julià Cufı́ et al. In this setting, the function ω 7→ fk (ω ) + 2(sin ω − ω ) plays a role; this function is R the sum of the functions of Hurwitz and Crofton. In spite of the fact that P ∈K / ( fk (ω ) + 2(sin ω − ω )) dP depends on k , the surprising fact is that, for k odd, decomposing the visual angle ω into the two parts ω1 , ω2 and adding the corresponding integrals leads to Z (fk (ω1 ) + 2(sin ω1 − ω1 ) + fk (ω2 ) + 2(sin ω2 − ω2 )) dP = 2πF, P ∈K / for each k ≥ 3 odd, as a consequence of Proposition 9. Moreover it will appear that the functions of Crofton and Hurwitz are in some sense a basis for the integral of any π -periodic or anti π -periodic function with respect to the density dG1 ∧ dG2 over the set of pairs of lines meeting a given compact convex Rset. More precisely, for every π -periodic or anti π -periodic function f , the integral G ∩K6=∅ f (ϕ(G2 ) − ϕ(G1 ))dG1 ∧ dG2 is a linear combination of i integrals extended outside K of Crofton and Hurwitz functions of the visual angle ω or of the two parts ω1 , ω2 of ω , respectively. 2 Densities in the space of ordered pairs of lines From now on we deal with the space of ordered pairs of lines endowed with the density dG1 ∧ dG2 . For every function f˜(G1 , G2 ) defined on this space we consider the new density f˜(G1 , G2 )dG1 ∧ dG2 . The measure of a set A of pairs of lines in the plane, provided A is measurable, is then given by Z f˜(G1 , G2 )dG1 ∧ dG2 . A We want now to characterize when this measure is invariant under Euclidean motions. Proposition 1 The density f˜(G1 , G2 )dG1 ∧ dG2 is invariant under the group of Eu- clidean motions if and only if f˜(G1 , G2 ) = f (ϕ(G2 ) −ϕ(G1 )) with f an even π-periodic function on R. Proof Let (pi , ϕi ) be the coordinates of Gi and define the function g by g (p1 , ϕ1 , p2 , ϕ2 ) = f˜(G1 , G2 ). The invariance of the density is equivalent to the equality g (p1 , ϕ1 , p2 , ϕ2 ) = g (p01 , ϕ01 , p02 , ϕ02 ) for each Euclidean motion sending the pair of lines with coordinates (p1 , ϕ1 , p2 , ϕ2 ) to the pair of lines with coordinates (p01 , ϕ01 , p02 , ϕ02 ). First of all let us show that g does not depend on p1 , p2 . In fact, for every straight line G = G(p, ϕ) and an arbitrary a > 0 there is a parallel line to G with coordinates (a, ϕ). Given two straight lines G1 = G(p1 , ϕ1 ), G2 = G(p2 , ϕ2 ) and two numbers a1 , a2 > 0 let G01 and G02 be the corresponding parallel lines with coordinates (a1 , ϕ1 ), (a2 , ϕ2 ). Performing the translation that sends the point G1 ∩ G2 to the point G01 ∩ G02 we have that g (p1 , ϕ1 , p2 , ϕ2 ) = g (a1 , ϕ1 , a2 , ϕ2 ) and so g does not depend on p1 and p2 . Given now the line G(p, ϕ) if we perform, for instance, the translation given by the vector −(p + )(cos ϕ, sin ϕ),  > 0, the translated line has coordinates (, ϕ + π ). Therefore the function g must be π -periodic with respect to the arguments ϕ1 , ϕ2 . Due to the invariance under rotations it follows that g (p1 , ϕ1 , p2 , ϕ2 ) = Integral Geometry about the visual angle of a convex set 5 g (p1 , 0, p2 , ϕ2 − ϕ1 ) and so g (p1 , ϕ1 , p2 , ϕ2 ) = f (ϕ2 − ϕ1 ) = f (ϕ(G2 ) − ϕ(G1 )) with f a π -periodic function. Finally the invariance under symmetries implies that f is an even function. Conversely it is clear that if f is an even π -periodic function then the density f (ϕ(G2 ) − ϕ(G1 ))dG1 ∧ dG2 is invariant under Euclidean motions. u t Our goal is now to integrate densities, not necessarily invariant under Euclidean motions, over the set of pairs of lines meeting a compact convex set K . We shall consider densities of the form f˜(G1 , G2 )dG1 ∧ dG2 = f (ϕ(G2 ) − ϕ(G1 ))dG1 ∧ dG2 with f a continuous 2π -periodic function. Notice that ϕ(G2 ) − ϕ(G1 ) gives one of the two angles between the lines G1 and G2 . In Theorem 1 we give a formula to compute the integral of the above densities in terms of both the Fourier coefficients of f and of the support function of K . Recall that when the origin of coordinates is an interior point of K , a hypothesis that we will assume from now on, the support function ϕ 7→ p(ϕ) is given by the distance to the origin of the tangent to K whose normal makes an angle ϕ with the positive part of the real axis (see [6]). Theorem 1 Let K be a compact convex set with C 1 boundary of length L. Let f be a 2π-periodic continuous function on R with Fourier expansion X f (ϕ) ∼ An cos(nϕ) + Bn sin(nϕ). n≥0 Then Z X f (ϕ(G2 ) − ϕ(G1 )) dG1 ∧ dG2 = A0 L2 + π 2 c2n An , (8) Gi ∩K6=∅ n≥1 with c2n = a2n + b2n where an , bn are the Fourier coefficients of the support function p of K. Proof Let (pi , ϕi ) be the coordinates of Gi . We have Z Z 2π Z 2π Z p(ϕ1 ) Z p(ϕ2 ) f (ϕ(G2 ) − ϕ(G1 )) dG1 ∧ dG2 = f (ϕ2 − ϕ1 )dp1 dp2 dϕ1 dϕ2 Gi ∩K6=∅ 0 0 0 0 Z 2π Z 2π = p(ϕ1 )p(ϕ2 )f (ϕ2 − ϕ1 ) dϕ1 dϕ2 . (9) 0 0 Performing the change of variables ϕ2 − ϕ1 = w, ϕ1 = u the integral (9) becomes Z 2π Z 2π−u p (u ) p(u + w)f (w) dw du. (10) 0 −u Since we are assuming that the boundary of K is of class C 1 the support function p is C 2 and so the Fourier development of p(u + w) in terms of the Fourier coefficients an , bn of p(u) is convergent and reads as X p(u + w) = a0 + (an cos(nu) + bn sin(nu)) cos(nw)+ n≥1  + (−an sin(nu) + bn cos(nu)) sin(nw) . 6 Julià Cufı́ et al. By the Plancherel identity the integral (10) is equal to Z 2π  X p(u) 2π A0 a0 + π An (an cos(nu) + bn sin(nu))+ 0 n≥1  + Bn (−an sin(nu) + bn cos(nu)) du = Z 2π  X  = p(u) 2π A0 a0 + π (An an + Bn bn ) cos(nu)+(An bn −Bn an ) sin(nu)) du = 0 n≥1 2 X = 4π A0 a20 + π2 (An an + Bn bn )an + (An bn − Bn an )bn = n≥1 X X = 4π 2 A0 a20 + π 2 An (a2n + b2n ) = A0 L2 + π 2 An c2n , n≥1 n≥1 where R 2π we have used that L = 2πa0 , which is 1a consequence of the equality L = 0 p(ϕ) dϕ (see for instance [6], Section I.1.2) , and the Theorem is proved. u t As it is well known (see [4]) the quantities c2k = a2k + b2k , k ≥ 2, are invariant under Euclidean motions of K . However c21 changes when moving K . So the integral in (8) is invariant under Euclidean motions of K if and only if A1 = 0. In particular this is the case when f is π -periodic. For a density given by a π -periodic function f and a compact set of constant width the measure of the pairs of lines that intersect K is proportional to L2 . More precisely we have Corollary 1 Let K be a compact convex set of constant width and f a continuous π-periodic function. Then Z f (ϕ(G2 ) − ϕ(G1 )) dG1 ∧ dG2 = λL2 , Gi ∩K6=∅ Rπ where λ = (1/π ) 0 f (ϕ) dϕ. Proof Since K is of constant width the Fourier development of p has only odd terms (see for instance §2 of [2]). Moreover the Fourier development of f has only even terms because it is a π -periodic function. Hence (8) gives Z f (ϕ(G2 ) − ϕ(G1 )) dG1 ∧ dG2 = A0 L2 , Gi ∩K6=∅ Rπ with A0 = (1/π ) 0 f (ϕ) dϕ. u t 1 Here one uses the hypothesis that the boundary of K is of class C 1 which implies that the support function p = p(ϕ) is C 2 . Then one has ds = (p + p00 )dϕ that integrating gives L = 02π p dϕ. R Integral Geometry about the visual angle of a convex set 7 3 Integral formulas of the visual angle in terms of densities in the space of pairs of lines In [2] there is a unified approach to several classical formulas involving integrals of functions of the visual angle of a compact convex set K . Among them one can find the integrals of Crofton, Masotti, powers of sine, and Hurwitz. The original proof of Crofton’s formula, via Integral Geometry, involves a mea- sure on the space of pairs of lines. The aim of this section is to interpret the formulas in [2] in terms of integrals of densities in the space of pairs of lines. To begin with let us consider Hurwitz’s formula Z fk (ω )dP = L2 + (−1)k π 2 (k2 − 1)c2k , (11) P ∈K / where fk is given in (3). For a proof of (11) see [4] or [2]. Comparing this equality with (8) one gets immediately the following result. Proposition 2 Let fk be the Hurwitz function defined in (3). Then Z Z   fk (ω )dP = 1 + (−1)k (k2 − 1) cos(k(ϕ(G2 ) − ϕ(G1 ))) dG1 ∧ dG2 . P ∈K / Gi ∩K6=∅ Nevertheless for the other quoted integral formulas it is not clear at all what density must be chosen. We shall provide a general method to find the densities corresponding to integrals of general functions of the visual angle. 3.1 A change of variables The classical proof of Crofton’s formula is based on the change of variables in the space of ordered pairs of lines given by (p1 , ϕ1 , p2 , ϕ2 ) −→ (P, α1 , α2 ) where P is the intersection point of the two straight lines and αi ∈ [0, π ] are the angles which determine the directions of the lines. More precisely the angle α associated to a line through a given point P is defined in the following way. Let − −→ u be a unitary vector orthogonal to OP where O is the origin of coordinates, and − −→ such that the basis (u, OP ) is positively oriented. Let G be a line through P with unitary director vector v such that the basis (u, v) is positively oriented. Then α = α(G) is defined by cos α = u · v and 0 < α < π . From now on we shall say that α is the direction of the line G. With these new coordinates, proceeding as in [6], Section I.4.3, one has dG1 ∧ dG2 = | sin(α2 − α1 )| dα1 ∧ dα2 ∧ dP. (12) We have used the fact that ϕ2 − ϕ1 = α2 − α1 + π where  = (P, α1 , α2 ) = 0, ±1, according to the position with respect to the origin of the lines G1 , G2 . As a consequence if f is a π -periodic function we have f (ϕ(G2 ) − ϕ(G1 )) dG1 ∧ dG2 = f (α2 − α1 )| sin(α2 − α1 )|dα1 ∧ dα2 ∧ dP. (13) 8 Julià Cufı́ et al. Fig. 1 Direction of a line. 3.2 Integrals of functions of pairs of lines meeting a convex set For a point P ∈ / K let α, β be the directions we have introduced corresponding to the support lines of K through P , with 0 < α < π/2 and π/2 < β < π . Then ω = β − α is the visual angle of K from P . This is the reason why we have slightly modified the definition of the direction angle given by Santaló in [6] as the angle between the line through P and the positive x axis, because with this definition one could have ω = β − α or ω = π − (β − α); see Figure 2. We shall provide now Fig. 2 Visual angle of a convex set. a general formula to calculate the integral of the right-hand side of (13). Proposition 3 Let f be a 2π-periodic continuous function on R, and H the C 2 func- tion on [−π, π ] satisfying the conditions H 00 (x) = f (x) sin(x), x ∈ [−π, π ], and H (0) = H 0 (0) = 0. Denote by αi the direction of the line Gi . Then Z f (α2 − α1 )| sin(α2 − α1 )| dα1 ∧ dα2 ∧ dP = Gi ∩K6=∅ Z = (H (π ) − H (−π ))F + (H (ω ) − H (−ω )) dP, P ∈K / Integral Geometry about the visual angle of a convex set 9 where ω = ω (P ) is the visual angle of K from P . Proof For a given point P in the plane there are angles α(P ), β (P ) such that the pairs of lines G1 , G2 through P that intersect the convex set K are those satisfying α(P ) ≤ αi ≤ β (P ), where αi = α(Gi ). When P ∈ K we have α(P ) = 0 and β (P ) = π . We need to integrate the function f (α2 − α1 )| sin(α2 − α1 )| over [α, β ]2 with α = α(P ) and β = β (P ). In order to perform this integral we divide [α, β ]2 into the union of the regions R1 = {(α1 , α2 ) ∈ [α, β ]2 : α2 ≥ α1 } and R2 = {(α1 , α2 ) ∈ [α, β ]2 : α2 < α1 }. Therefore Z f (α2 − α1 )| sin(α2 − α1 )| dα1 dα2 = [α,β ]2 Z Z = f (α2 − α1 ) sin(α2 − α1 ) dα1 dα2 − f (α2 − α1 ) sin(α2 − α1 ) dα1 dα2 = R1 R2 Z β Z α2  = f (α2 − α1 ) sin(α2 − α1 )dα1 dα2 α α Z β Z α1  − f (α2 − α1 ) sin(α2 − α1 )dα2 dα1 = α α Z β α2 Z β α1 −H 0 (α2 − α1 ) H 0 (α2 − α1 )   = α dα2 − α dα1 = α α = [H (α2 − α)]βα − [H (α − α1 )]βα = H (β − α) − H (α − β ). Hence Z f (α2 − α1 )| sin(α2 − α1 )| dα1 ∧ dα2 ∧ dP = Gi ∩K6=∅ Z Z  = + (H (β − α) − H (α − β )) dP. P ∈K P ∈K / Taking into account that the visual angle ω (P ) is given by β (P ) − α(P ) the result follows. u t In the next result we assume the additional hypothesis that f is an even func- tion. Proposition 4 Let f be a 2π-periodic continuous function on R, with f (−x) = f (x), x ∈ R, and let H be the C 2 function on [0, π ] satisfying the conditions H 00 (x) = f (x) sin(x), x ∈ [0, π ], and H (0) = H 0 (0) = 0. Denote by αi the direction of the line Gi . Then Z Z f (α2 − α1 )| sin(α2 − α1 )| dα1 ∧ dα2 ∧ dP = 2H (π )F + 2 H (ω ) dP, Gi ∩K6=∅ P ∈K / where ω = ω (P ) is the visual angle of K from P . 10 Julià Cufı́ et al. Proof Just proceed as in the above proof taking into account that Z f (α2 − α1 )| sin(α2 − α1 )| dα1 dα2 = [α,β ]2 Z β Z α2  =2 f (α2 − α1 ) sin(α2 − α1 )dα1 dα2 . α α u t For the special case where f is a π -periodic function one has Corollary 2 Let f be a π-periodic continuous function on R, and let H be the C 2 function on [−π, π ] satisfying the conditions H 00 (x) = f (x) · sin(x), x ∈ [−π, π ], and H (0) = H 0 (0) = 0. Then Z f (ϕ(G2 ) − ϕ(G1 )) dG1 ∧ dG2 = Gi ∩K6=∅  Z  = (H (π ) − H (−π ))F + (H (ω ) − H (−ω )) dP . P ∈K / If moreover f (−x) = f (x) and H is C 2 on [0, π ] with H 00 (x) = f (x) · sin(x), x ∈ [0, π ], and H (0) = H 0 (0) = 0, one has Z Z f (ϕ(G2 ) − ϕ(G1 )) dG1 ∧ dG2 = 2H (π )F + 2 H (ω ) dP. (14) Gi ∩K6=∅ P ∈K / Proof When f is a π -periodic function we have equality (13) and the result is then a consequence of Proposition (3) and Proposition (4). u t Remark that according to Proposition 1, equality (14) holds for densities that are invariant under Euclidean motions. Integral formulas as those given in (8) and (14) open the possibility to prove interesting relations for quantities linked to convex sets. For instance when applied to the function f (x) = cos(kx) they give Hurwitz’s formula (11) for k even (see section 4). For odd values of k the Corollary 2 does not apply because f (x) = cos(kx) is not a π -periodic function. In this case we have f (x + π ) = −f (x) and we say that f is an anti π-periodic function. For this type of functions we can modify the above proofs to obtain a new result that involves a decomposition of the visual angle ω into ω = ω1 + ω2 where ω1 and ω2 are defined in the following way. Given a point P ∈/ K we have considered in section 3.2 the directions 0 < α < π/2 < β < π of the support lines of K through P and the visual angle ω = β − α. Let us take ω1 = π/2 − α and ω2 = β − π/2. Then we have Proposition 5 Let f be an anti π-periodic continuous function on R such that f (x) = f (−x) and let H be the C 2 function on [0, π ] with H 00 (x) = f (x) sin(x), x ∈ [0, π ], and H (0) = H 0 (0) = 0. Then Z f (ϕ(G2 ) − ϕ(G1 )) dG1 ∧ dG2 = Gi ∩K6=∅ Z = 2(2H (π/2) − H (π ))F + 2 (2H (ω1 ) + 2H (ω2 ) − H (ω )) dP. (15) P ∈K / Integral Geometry about the visual angle of a convex set 11 Fig. 3 Angles ω1 and ω2 . Proof In Section 3.1 we have seen that ϕ2 − ϕ1 = α2 − α1 + π where  = (P, α1 , α2 ) = 0, ±1. Then Z f (ϕ(G2 ) − ϕ(G1 )) dG1 ∧ dG2 = Gi ∩K6=∅ Z Z = (−1) f (α2 − α1 )| sin(α2 − α1 )| dα1 dα2 dP. P ∈R2 [α(P ),β (P )]2 If P ∈ / K we consider the regions R1 = {(α1 , α2 ) : α ≤ α1 < α2 ≤ π/2} R2 = {(α1 , α2 ) : π/2 ≤ α1 < α2 ≤ β} R3 = {(α1 , α2 ) : α ≤ α1 < π/2 < α2 ≤ β}. In R1 and R2 we have  = 1 and  = −1 in region R3 . Therefore, for P ∈ /K Z β Z β (−1) f (α2 − α1 )| sin(α2 − α1 )| dα1 dα2 = α Z α Z  =2 f (α2 − α1 ) sin(α2 − α1 ) dα1 dα2 + f (α2 − α1 ) sin(α2 − α1 ) dα1 dα2 R1 R Z 2 =− f (α2 − α1 ) sin(α2 − α1 ) dα1 dα2 R3 12 Julià Cufı́ et al. The integrals over R1 and R2 are easily computed and their values are 2H (ω1 ) and 2H (ω2 ) respectively. Let us compute the third integral. Z f (α2 − α1 ) sin(α2 − α1 ) dα1 dα2 = R3 Z β Z π/2 = f (α2 − α1 ) sin(α2 − α1 ) dα1 dα2 = π/2 α Z β α1 =π/2 Z β −H 0 (α2 − α1 ) (H 0 (α2 − α) − H 0 (α2 − π/2))dα2 =  =2 α1 =α dα2 = 2 π/2 π/2 = 2 [H (α2 − α) − H (α2 − π/2)]α 2 =β α2 =π/2 = 2 (H (ω ) − H (β − π/2) − H (π/2 − α)) = = 2 (H (ω ) − H (ω2 ) − H (ω1 )) . Finally, for G1 ∩ G2 = P ∈ / K we have Z Z f (ϕ(G2 ) − ϕ(G1 )) dG1 ∧ dG2 = 2 (2H (ω1 ) + 2H (ω2 ) − H (ω )) dP. (16) P ∈K / P ∈K / When P ∈ K we do the same computations but now α = 0, β = π and ω = β−α = π and so ω1 = π/2 = ω2 . Thus Z f (ϕ(G2 ) − ϕ(G1 )) dG1 ∧ dG2 = 2(4H (π/2) − H (π ))F. (17) G1 ∩G2 ∈K Joining (16) and (17) the Proposition follows. u t 3.3 Interpretation in terms of densities of the formulas of Crofton, Masotti and powers of sine In this section we will give an interpretation of the integrals of the visual angle appearing in the formulas of Crofton, Masotti and power of sine in terms of inte- grals of densities in the space of pairs of lines. All these densities will be given by even π -periodic functions and so they will be invariant under Euclidean motions. For Hurwitz’s formula this was done in Proposition 2. Crofton’s formula Taking H (x) = x − sin(x) it follows that f = 1 in Corollary 2 and since H (π ) = π using (14) we get Proposition 6 The following equality holds. Z Z dG1 ∧ dG2 = 2πF + 2 (ω − sin ω ) dP. Gi ∩K6=∅ P ∈K / Integral Geometry about the visual angle of a convex set 13 Masotti’s formula Taking H (x) = x2 − sin2 (x) one gets H 00 (x)/ sin(x) = 4 sin(x). So the function f (x) = 4| sin(x)|, x ∈ R, satisfies the hypothesis of Corollary 2 and equation (14) gives Proposition 7 The following equality holds Z Z 2 | sin(ϕ(G2 ) − ϕ(G1 ))|dG1 ∧ dG2 = π 2 F + (ω 2 − sin2 ω ) dP. Gi ∩K6=∅ P ∈K / Powers of sine formula Finally, in an analogous way we can interpret the integral of any power of the sine of the visual angle. Effectively for H (x) = sinm (x) it follows that H 00 (x)/ sin(x) = m(m − 1) sinm−3 (x) − m2 sinm−1 (x). So taking f (x) = m(m− 1)| sinn−3 (x)|−m2 | sinm−1 (x)| the hypotheses of Corollary 2 are satisfied and by (14) we have Proposition 8 The following equality holds Z 2 sinm (ω ) dP = P ∈K / Z   = m(m − 1)| sinm−3 (ϕ(G2 ) − ϕ(G1 ))| − m2 | sinm−1 (ϕ(G2 ) − ϕ(G1 ))| dG1 ∧dG2 . Gi ∩K6=∅ 4 New proofs of classical formulas Combining the results of the previous section with Theorem 1 new proofs of the formulas of Masotti and the powers of sine can be obtained, in the spirit of the classical proof of Crofton’s formula via Integral Geometry R To begin with we note that Theorem 1 implies the equality Gi ∩K6=∅ dG1 ∧ dG2 = L2 which is also an immediate consequence of the well known Cauchy- Crofton’s formula (see [6], Section I.4.3). Now this equality together with Propo- sition 6 gives Crofton’s formula Z L2 = 2πF + 2 (ω − sin ω ) dP. (18) P ∈K / Masotti’s formula A simple calculation shows that the Fourier expansion of the function | sin(t)| is 2 4 X cos(2nt) | sin(t)| = + . (19) π π 1 − 4n2 n≥1 14 Julià Cufı́ et al. So by Theorem 1, 2L2 c22n Z X | sin(ϕ(G2 ) − ϕ(G1 ))| dG1 ∧ dG2 = + 4π , Gi ∩K6=∅ π 1 − 4n2 n≥1 and using Proposition 7 one gets 4L2 c22n Z X (ω 2 − sin2 ω ) dP = −π 2 F + + 8π , P ∈K / π 1 − 4n2 n≥1 which is Masotti’s formula (5). Another example In the preceding sections we have interpreted integral formulas of some functions of the visual angle in terms of densities in the space of pairs of lines. But one can also proceed in the reverse sense, that is to start from a density and to look for the corresponding function of the visual angle. R For instance the proof of Masotti’s formula leads to compute Gi ∩K | sin(ϕ(G2 )− ϕ(G1 ))| dG1 ∧ dG2 . If we consider now the function | cos(ϕ(G2 ) − ϕ(G1 ))|, using Theorem 1 and that 2 4 X (−1)n cos(2nt) | cos(t)| = + π π 1 − 4n2 n≥1 we get Z 2L2 X (−1)n c2 2n | cos(ϕ(G2 ) − ϕ(G1 ))|dG1 ∧ dG2 = + 4π . Gi ∩K6=∅ π 1 − 4n2 n≥1 The function H appearing in Corollary 2 is in this case (1 4 (ω − sin ω cos ω ) 0 ≤ ω ≤ π/2 H (ω ) = 1 4 (3ω − π + sin ω cos ω ) π/2 ≤ ω ≤ π. Hence, by (14) we have Z Z | cos(ϕ(G2 ) − ϕ(G1 ))|dG1 ∧ dG2 = πF + 2 H (ω )dP. Gi ∩K6=∅ P ∈K / Powers of sine formula In order to apply Theorem 1 to the right-hand side of the equality in Proposition 8 we need to compute the Fourier coefficients of the function f (x) = m(m − 1)| sinm−3 (x)| − m2 | sinm−1 (x)|. It is clear that Ak = 0 for k odd. For k even we have 1 2π Z Ak = f (x) cos(kx)dx = π 0 Z π Z π 1 = [2m(m − 1) sinm−3 x cos(kx) dx − 2m2 sinm−1 x cos(kx) dx] = π 0 0 1 = [2m(m − 1)Im−3,k − 2m2 Im−1,k ] (20) π Integral Geometry about the visual angle of a convex set 15 where π 2−m m!π Z Im,k = sinm (x) cos(kx) dx = (−1)k/2 m−k m+k , 0 Γ (1 + 2 )Γ (1 + 2 ) (see, for instance, [3], p. 372). Substituting this expression in (20) it follows k m! (−1) 2 +1 (k2 − 1) Ak = . 2m−2 (m − 2) Γ ( m+1+k )Γ ( m+1−k ) 2 2 Finally using Theorem 1 we get Z m! sinm (ω ) dP = L2 P ∈K / 2m (m − 2)Γ ( m− 2 1 2 ) k m!π 2 X (−1) 2 +1 (k2 − 1) + m− m+1−k k c2 . 2 1 (m − 2) Γ ( m+1+k 2 ) Γ ( 2 ) k≥2,even Note that for m odd the index k in the sum runs only from 2 to m − 1. This formula, which was first obtained by a different method in [2], provides an interpretation of the coefficients of c2k as the Fourier coefficients of the above function f . Crofton-Hurwitz’s integral In the above two previous sections we have strongly used equality (14) of Corollary 2, which depends on the fact that the function f is π -periodic, a fact that is crucial in order that equality (13) holds. Consider now the function f (x) = cos(kx) with k > 1. This function satisfies the hypothesis of Corollary 2 for k even and the hypothesis of Proposition 5 for k odd. We have that 1 Hk (x) = (f (x) + 2(sin x − x)) , (21) 2(k2 − 1) k with fk the Hurwitz’s function given in (3), satisfies the equation Hk00 (x) = cos(kx) · sin x, x ∈ [0, π ], and Hk (0) = Hk0 (0) = 0. Therefore, for k even, equalities (8) and (14) give Z Z πF π 2 c2k = cos(k(ϕ(G2 ) − ϕ(G1 ))) dG1 ∧ dG2 = − 2 +2 Hk (ω ) dP, Gi ∩K6=∅ k −1 P ∈K / and using Crofton’s formula (18) one gets a new proof of Hurwitz’s formula (11) for k even. When k is odd equation (15) gives Z cos(k(ϕ(G2 ) − ϕ(G1 ))) dG1 ∧ dG2 = Gi ∩K6=∅ Z 2πF =− 2 +2 (2Hk (ω1 ) + 2Hk (ω2 ) − Hk (ω )) dP. k −1 P ∈K / 16 Julià Cufı́ et al. Using the equality (8) one deduces that π 2 c2k Z Z 2πF Hk (ω )dP = − − 2 + (Hk (ω1 ) + Hk (ω2 )) dP. P ∈K / 2 k −1 P ∈K / Now by (21) and Crofton’s formula we obtain Z Z fk (ω )dP = L2 − π 2 (k2 − 1)c2k − 2πF + 2(k2 − 1) (Hk (ω1 ) + Hk (ω2 ))dP. P ∈K / P ∈K / R (22) Since we do not know the value of P ∈K / ( H k ( ω1 ) + H k ( ω2 )) dP we are not able to prove Hurwitz’s formula in the case of k odd. But from (11) we get the following result. Proposition 9 Let K be a compact convex set of area F . Then Z (k2 − 1) (Hk (ω1 ) + Hk (ω2 )dP = πF (23) P ∈K / for each k ≥ 3 odd, where Hk is given in (21). Notice that the above equation is equivalent to Z (fk (ω1 ) + 2(sin ω1 − ω1 ) + fk (ω2 ) + 2(sin ω2 − ω2 )) dP = 2πF. (24) P ∈K / The function Hk is the sum, except for a constant, of Hurwitz’s function and Crofton’s function and so are the terms in the above integrand. The integral of the sum of Crofton’s and Hurwitz’s functions of the visual angle is Z (fk (ω ) + 2(sin ω − ω ))dP = 2πF + (−1)k π 2 (k2 − 1)c2k , k ≥ 2. P ∈K / The surprising fact is that, for k odd, decomposing the visual angle ω into the two parts ω = ω1 + ω2 and adding the corresponding integrals one gets (24) in which the right-hand side does not depend on k. R In concluding we make the following remark. Theorem 1 states that the integral Gi ∩K6=∅ f (ϕ(G2 ) − ϕ(G1 )) dG1 ∧ dG2 depends only on the integrals Z cos(k(ϕ(G2 ) − ϕ(G1 ))) dG1 ∧ dG2 . Gi ∩K6=∅ So, by the results of section 3.1 we are led to calculate the functions Hk (x) such that Hk00 (x) = cos(kx) sin(x) with Hk (0) = Hk0 (0) = 0. These functions appear to be the sum of the functions of Hurwitz and Crofton given in (21), that is 1 H k (x ) = (f (x) + 2(sin x − x)) , k ≥ 2, 2(k2 − 1) k and H1 (x) = (1/8)(2x − sin(2x)). As a Rconsequence when f is a π -periodic function, according to Corollary 2, the integral Gi ∩K6=∅ f (ϕ(G2 ) − ϕ(G1 )) dG1 ∧ dG2 is a linear combination of integrals extended outside K of the functions of the visual angle Hk (ω ). Likewise when the function f is anti π -periodic, according to Proposition 5, the corresponding Integral Geometry about the visual angle of a convex set 17 integral is a linear combination of integrals extended outside K of the functions Hk (ω ), Hk (ω1 ) and Hk (ω2 ). Summarizing, it appears that the functions of Crofton and Hurwitz are some kind of basis for the integral of any π -periodic or anti π -periodic function with respect to the density dG1 ∧ dG2 over the set of pairs of lines meeting a given compact convex set. References 1. M. W. Crofton. On the theory of local probability. Phil. Trans. R. Soc. Lond., 158:181–199, 1868. 2. J. Cufı́, E. Gallego, and A. Reventós. On the integral formulas of Crofton and Hurwitz relative to the visual angle of a convex set. Mathematika, 65:874–896, 2019. 3. I. S. Gradshteyn and I. M. Ryzhik. Table of integrals, series, and products. Academic Press, New York-London-Toronto, Ont., 1980. 4. A. Hurwitz. Sur quelques applications géométriques des séries de Fourier. Annales scien- tifiques de l’É.N.S., 3ème série, 19:357–408, dec 1902. 5. G. Masotti Biggiogero. La Geometria Integrale. Rend. Sem. Mat. Fis. Milano, 25:164–231 (1955), 1953–54. 6. L. A. Santaló. Integral geometry and geometric probability. 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