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Crofton formula

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In mathematics, the Crofton formula, named after Morgan Crofton (1826–1915), (also Cauchy-Crofton formula) is a classic result of integral geometry relating the length of a curve to the expected number of times a "random" line intersects it.

Statement[edit]

The line defined by choices of intersects the curve twice, therefore, .
Application of the Crofton formula in a Monte-Carlo simulation.

Suppose is a rectifiable plane curve. Given an oriented line , let () be the number of points at which and intersect. We can parametrize the general line by the direction in which it points and its signed distance from the origin. The Crofton formula expresses the arc length of the curve in terms of an integral over the space of all oriented lines:

The differential form

is invariant under rigid motions of , so it is a natural integration measure for speaking of an "average" number of intersections. It is usually called the kinematic measure.

The right-hand side in the Crofton formula is sometimes called the Favard length.[1]

In general, the space of oriented lines in is the tangent bundle of , and we can similarly define a kinematic measure on it, which is also invariant under rigid motions of . Then for any rectifiable surface of codimension 1, we have where

Proof sketch[edit]

Both sides of the Crofton formula are additive over concatenation of curves, so it suffices to prove the formula for a single line segment. Since the right-hand side does not depend on the positioning of the line segment, it must equal some function of the segment's length. Because, again, the formula is additive over concatenation of line segments, the integral must be a constant times the length of the line segment. It remains only to determine the factor of 1/4; this is easily done by computing both sides when γ is the unit circle.

The proof for the generalized version proceeds exactly as above.

Poincare’s formula for intersecting curves[edit]

Let be the Euclidean group on the plane. It can be parametrized as , such that each defines some : rotate by counterclockwise around the origin, then translate by . Then is invariant under action of on itself, thus we obtained a kinematic measure on .

Given rectifiable simple (no self-intersection) curves in the plane, then The proof is done similarly as above. First note that both sides of the formula are additive in , thus the formula is correct with an undetermined multiplicative constant. Then explicitly calculate this constant, using the simplest possible case: two circles of radius 1.

Other forms[edit]

The space of oriented lines is a double cover of the space of unoriented lines. The Crofton formula is often stated in terms of the corresponding density in the latter space, in which the numerical factor is not 1/4 but 1/2. Since a convex curve intersects almost every line either twice or not at all, the unoriented Crofton formula for convex curves can be stated without numerical factors: the measure of the set of straight lines which intersect a convex curve is equal to its length.

The same formula (with the same multiplicative constants) apply for hyperbolic spaces and spherical spaces, when the kinematic measure is suitably scaled. The proof is essentially the same.

The Crofton formula generalizes to any Riemannian surface or more generally to two-dimensional Finsler manifolds; the integral is then performed with the natural measure on the space of geodesics.[2]

More general forms exist, such as the kinematic formula of Chern.[3]

Applications[edit]

Crofton's formula yields elegant proofs of the following results, among others:

  • Given two nested, convex, closed curves, the inner one is shorter. In general, for two such codimension 1 surfaces, the inner one has less area.
  • Given two nested, convex, closed surfaces , with nested inside , the probability of a random line intersecting the inner surface , conditional on it intersecting the outer surface , is This is the justification for the surface area heuristic in bounding volume hierarchy.
  • Given compact convex subset , let be a random line, and be a random hyperplane, then where is the average width of , that is, the expected length of the orthogonal projection of to a random linear subspace of . When , by the isoperimetric inequality, this probability is upper bounded by , with equality iff is a disk.
  • Barbier's theorem: Every curve of constant width w has perimeter πw.
  • The isoperimetric inequality: Among all closed curves with a given perimeter, the circle has the unique maximum area.
  • The convex hull of every bounded rectifiable closed curve C has perimeter at most the length of C, with equality only when C is already a convex curve.
  • Cauchy's surface area formula: Given any convex compact subset , let be the expected shadow area of (that is, is the orthogonal projection to a random hyperplane of ), then by integrating Crofton formula first over , then over , we getIn particular, setting gives Barbier's theorem, gives the classic example "the average shadow of a convex body is 1/4 of its surface area". General gives generalization of Barbier's theorem for bodies of constant brightness.

See also[edit]

References[edit]

  1. ^ Luis Santaló (1976), Integral geometry and geometric probability, Addison-Wesley, ISBN 0-201-13500-0
  2. ^ Ueno, Seitarô (1955), "On the densities in a two-dimensional generalized space", Memoirs of the Faculty of Science, 9: 65–77, doi:10.2206/kyushumfs.9.65, MR 0071801
  3. ^ Calegari, Danny (2020). "On the Kinematic Formula in the Lives of the Saints" (PDF). Notices of the American Mathematical Society. 67 (7): 1042–1044. ISSN 0002-9920. Archived from the original (PDF) on 20 November 2020. Retrieved 7 June 2022.
  4. ^ Izrail Moiseevich Gel'fand; Mark Iosifovich Graev (1991), "Crofton's function and inversion formulas in real integral geometry", Functional Analysis and Its Applications, 25: 1–5, doi:10.1007/BF01090671, S2CID 24484682

External links[edit]