Crofton formula

File:CroftonFormula.png

File:Crofton formula multiple lines.png.]]

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

:$\backslash operatorname\{length\}\; (\backslash gamma)\; =\; \backslash frac14\backslash iint\; n\_\backslash gamma(\backslash varphi,\; p)\backslash ;\; d\backslash varphi\backslash ;\; dp.$

The differential form

:$d\backslash varphi\backslash wedge\; dp$

is invariant under rigid motions of $\backslash R^2$, 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.{{citation|author1=Luis Santaló|title=Integral geometry and geometric probability|publisher=Addison-Wesley|year=1976 |isbn=0-201-13500-0}}

In general, the space of oriented lines in $\backslash R^n$ is the tangent bundle of $S^\{n-1\}$, and we can similarly define a kinematic measure $d\backslash varphi\backslash wedge\; dp$ on it, which is also invariant under rigid motions of $\backslash R^n$. Then for any rectifiable surface $S$ of codimension 1, we have $$\backslash operatorname\{area\}\; (S)\; =\; C\_n\; \backslash iint\; n\_\backslash gamma(\backslash varphi,\; p)\backslash ;\; d\backslash varphi\backslash ;\; dp.$$where$$C\_n\; =\; \backslash frac\{1\}\{2\; \backslash cdot\; |\backslash text\{unit\; ball\; in\; \}\backslash R^\{n-1\}|\}\; =\; \backslash frac\{\backslash Gamma\{(\backslash frac\{n+1\}\{2\})\}\}\{2\backslash pi^\{\backslash frac\{n-1\}\{2\}\}\}$$

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.

Let $E^2$ be the Euclidean group on the plane. It can be parametrized as $[0,\; 2\backslash pi)\; \backslash times\; \backslash R^2$, such that each $(\backslash varphi,\; x,\; y)\backslash in\; [0,\; 2\backslash pi)\; \backslash times\; \backslash R^2$ defines some $T(\backslash varphi,\; x,\; y)$: rotate by $\backslash varphi$ counterclockwise around the origin, then translate by $(x,\; y)$. Then $dx\backslash wedge\; dy\; \backslash wedge\; d\backslash varphi$ is invariant under action of $E^2$ on itself, thus we obtained a kinematic measure on $E^2$.

Given rectifiable simple (no self-intersection) curves $C,\; D$ in the plane, then $$\backslash int\_\{T\backslash in\; E^2\}\; |C\backslash cap\; T(D)|\; dT\; =\; 4|C|\backslash cdot\; |D|$$The proof is done similarly as above. First note that both sides of the formula are additive in $C,\; D$, 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.

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.{{citation

| last = Ueno | first = Seitarô

| doi = 10.2206/kyushumfs.9.65

| journal = Memoirs of the Faculty of Science

| mr = 71801

| pages = 65–77

| title = On the densities in a two-dimensional generalized space

| volume = 9

| year = 1955}}

More general forms exist, such as the kinematic formula of Chern.{{Cite journal |last=Calegari |first=Danny |date=2020 |title=On the Kinematic Formula in the Lives of the Saints |url=https://www.ams.org/notices/202007/rnoti-p1042.pdf |journal=Notices of the American Mathematical Society |volume=67 |issue=7 |pages=1042–1044 |issn=0002-9920 |archive-url=https://web.archive.org/web/20201120195857/https://www.ams.org/notices/202007/rnoti-p1042.pdf |archive-date=20 November 2020 |url-status=dead |access-date=7 June 2022 }}

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 $S\_1,\; S\_2$, with $S\_1$ nested inside $S\_2$, the probability of a random line $l$ intersecting the inner surface $S\_1$, conditional on it intersecting the outer surface $S\_2$, is$$Pr(l\; \backslash text\{\; intersects\; \}\; S\_1|l\; \backslash text\{\; intersects\; \}\; S\_2)\; =\; \backslash frac\{\backslash operatorname\{area\}(S\_1)\}\{\backslash operatorname\{area\}(S\_2)\}$$ This is the justification for the surface area heuristic in bounding volume hierarchy.
• Given compact convex subset $S\backslash subset\backslash R^n$, let $l$ be a random line, and $P$ be a random hyperplane, then $$Pr(l\backslash text\{\; intersects\; \}\; P|\; l,\; P\backslash text\{\; intersects\; \}\; S)\; =\; \backslash frac$$

  S

  \partial S| \cdot E[\text{width of }S]}where $E[\backslash text\{width\; of\; \}S]$ is the average width of $S$, that is, the expected length of the orthogonal projection of $S$ to a random linear subspace of $\backslash R^n$. When $n=2$, by the isoperimetric inequality, this probability is upper bounded by $\backslash frac\; 1\; 2$, with equality iff $S$ is a disk. Barbier's theorem: Every curve of constant width w has perimeter {{pi}}w. 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 $S\backslash subset\; \backslash R^n$, let $E[|T(S)|]$ be the expected shadow area of $S$ (that is, $T$ is the orthogonal projection to a random hyperplane of $\backslash R^n$), then by integrating Crofton formula first over $dp$, then over $d\backslash varphi$, we get$$\backslash frac\{|\backslash partial\; S$$

  {E[|T(S)|]}

= \frac

= 2\sqrt\pi\frac{\Gamma(\frac{n+1}{2})}{\Gamma(\frac{n}{2})}In particular, setting $n=2$ gives Barbier's theorem, $n=3$ gives the classic example "the average shadow of a convex body is 1/4 of its surface area". General $n$ gives generalization of Barbier's theorem for bodies of constant brightness.

• Buffon's noodle
• The Radon transform can be viewed as a measure-theoretic generalization of the Crofton formula and the Crofton formula is used in the inversion formula of the k-plane Radon transform of Gel'fand and Graev {{citation|author1= Izrail Moiseevich Gel'fand|author2= Mark Iosifovich Graev|title=Crofton's function and inversion formulas in real integral geometry|journal=Functional Analysis and Its Applications|year=1991 |volume= 25|pages= 1–5|doi=10.1007/BF01090671|s2cid= 24484682}}
• Steinhaus longimeter

{{Reflist}}

• {{cite book | first=Serge | last=Tabachnikov | authorlink=Sergei Tabachnikov| year=2005 | title=Geometry and Billiards | publisher=AMS | pages=36–40 | isbn=0-8218-3919-5}}
• {{cite book | first=L. A. | last=Santalo | year=1953 | title=Introduction to Integral Geometry | pages=12–13, 54 | id={{LCC|QA641.S3}}}}

• [http://merganser.math.gvsu.edu/david/reed03/projects/weyhaupt/project.html Cauchy–Crofton formula page], with demonstration applets
• [https://www.youtube.com/watch?v=ltLUadnCyi0&ab_channel=3Blue1Brown Alice, Bob, and the average shadow of a cube], a visualization of Cauchy's surface area formula.

Category:Integral geometry

Category:Measure theory

Category:Differential geometry