VII. On the theory of local probability, applied to straight lines drawn at random in a plane; the methods used being also extended to the proof of certain new theorems in the integral calculus

Abstract

On the Theory o f Local Probability, applied to Straight Lines drawn at random in a plane ; the methods used being also extended to the proof o f certain new Theorems in the Integral C a l c u l u s. B y M organ W. Crofton, B.A., o f the Military Academy, Woolwich ; late Professor o f Natural Philosophy in the Queen's University, Ireland. Communicated by J. J.

Reaction Rates in Chemically Heterogeneous Rock: Coupled Impact of Structure and Flow Properties Studied by X-ray Microtomography

“…Figure S5 shows the plot of the porosity profile of this subvolume (A) and the effective perimeter of the porosity per image slice with time (B). The perimeter was calculated for every 2D image slice perpendicular to the direction of flow using the Crofton formula, which counts the number of image line intersections to estimate curve lengths. The porosity profile increases evenly across the length of the core, suggesting that no wormholing occurs but that channel expansion does.…”

New rotational integrals in space forms, with an application to surface area estimation

“…Further, let L n r denote an r-plane (r n), namely a totally geodesic submanifold of dimension r in M , respectively ( [13], [12]). One of the first problems considered in integral geometry was to determine the densities dL 1 was proposed by Crofton (see [4]), whereas the invariant densities for lines and planes in M 3 λ were obtained by Cartan [3] in 1896. In 1935, Blaschke [2] introduced the density dL n r for R n .…”

Early History of Geometric Probability and Stereology

“…After the proof of the Crofton formula Eq. 21, Czuber recalls another result contained in (Crofton, 1868), namely that the probability that a line hitting a closed convex curve L of length L hits also a closed convex curve ℓ of length l that lies inside L is p = l/L. Then he remarks that this result provides an experimental rectification of a closed convex curve:…”

“…Fréchet supplemented Hostinský's ideas in Fréchet (1921); see also Fréchet and Halbwachs (1924). Five years later Hostinský published a French booklet (Hostinský, 1925), in which he extended the contributions of Crofton (1868) and Czuber (1884a; for surfaces in space. Besides other results he proved the analogy of the Crofton-Hostinský formula mentioned above, which is probably the reason why Hostinský's name has been associated with the name of Crofton in this connection.…”

“…7. 18 From other problems considered by Crofton (1868), let us at least mention the generalization of Buffon's problem to a "needle" consisting of two rigidly connected figures of diameters not exceeding the distance between parallel lines, later extended by Sylvester (1890) to an arbitrarily long chain of such figures.…”