Integral geometry about the visual angle of a convex set
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Integral geometry about the visual angle of a convex set Julià Cufí1 · Eduardo Gallego1
· Agustí Reventós1
Received: 18 June 2019 / Accepted: 9 October 2019 / Published online: 15 October 2019 © Springer-Verlag Italia S.r.l., part of Springer Nature 2019
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 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 2 (ω − sin ω)d P + 2π F = L 2 , (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
The authors were partially supported by grants 2017SGR358, 2017SGR1725 (Generalitat de Catalunya) and PGC2018-095998-B-100 (Ministerio de Economía y Competitividad).
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Eduardo Gallego [email protected] Julià Cufí [email protected] Agustí Reventós [email protected]
1
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain
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of all pairs of lines meeting K is L 2 , twice the integral of ω − sin ω with respect to the area element d P is the measure of those pairs of lines intersecting themselves outside K and 2π F 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 f k (ω)d P = L 2 + (−1)k π 2 (k 2 − 1)ck2 , (2) P ∈K /
where f k (ω) = −2 sin ω +
k+1 k−1 sin((k − 1)ω) − sin((k + 1)ω), k ≥ 2, k−1 k+1
and ck2 = ak2 + bk2 , with ak , bk the Fourier coefficients of the support function of K . In the particular case k = 2 formula (2) gives 3 9 sin3 ω d P = L 2 + π 2 c22 . 4 4 P ∈K /
(3)
(4)
Masotti Biggiogero in [5] states without proof the following Crofton’s type formula 1 4L 2 2 2 2 (ω − sin ω) d P = −π F + (5) + 8π c2 . π 1 − 4k 2 2k P ∈K / 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: m! sinm (ω) d P = L2 m−1 2 m 2 (m − 2)Γ ( ) P ∈K / 2 +
m!π 2 m−1 2 (m − 2)
(−1) 2 +1 (k 2 − 1)
k≥2,even
Γ ( m+1+k
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