In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Crofton formula, named after
Morgan Crofton
Morgan Crofton (1826, Dublin, Ireland – 1915, Brighton, England) was an Irish mathematician who contributed to the field of geometric probability theory. He also worked with James Joseph Sylvester and contributed an article on probability to ...
(1826–1915), is a classic result of
integral geometry In mathematics, integral geometry is the theory of measures on a geometrical space invariant under the symmetry group of that space. In more recent times, the meaning has been broadened to include a view of invariant (or equivariant) transformation ...
relating the length of a curve to the
expected number of times a "random"
line
Line most often refers to:
* Line (geometry), object with zero thickness and curvature that stretches to infinity
* Telephone line, a single-user circuit on a telephone communication system
Line, lines, The Line, or LINE may also refer to:
Arts ...
intersects it.
Statement
Suppose
is a
rectifiable plane curve
In mathematics, a plane curve is a curve in a plane that may be either a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic pla ...
. Given an oriented line ''ℓ'', let
(''ℓ'') be the number of points at which
and ''ℓ'' intersect. We can parametrize the general line ''ℓ'' by the direction
in which it points and its signed distance
from the
origin
Origin(s) or The Origin may refer to:
Arts, entertainment, and media
Comics and manga
* Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002
* The Origin (Buffy comic), ''The Origin'' (Bu ...
. The Crofton formula expresses the
arc length
ARC may refer to:
Business
* Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s
* Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services
* ...
of the curve
in terms of an
integral
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
over the space of all oriented lines:
:
The
differential form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
:
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.
In general, the space of oriented lines in
is the
tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
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
Both sides of the Crofton formula are
additive
Additive may refer to:
Mathematics
* Additive function, a function in number theory
* Additive map, a function that preserves the addition operation
* Additive set-functionn see Sigma additivity
* Additive category, a preadditive category with f ...
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
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucl ...
.
The proof for the generalized version proceeds exactly as above.
Poincare’s formula for intersecting curves
Let
be the
Euclidean group
In mathematics, a Euclidean group is the group of (Euclidean) isometries of a Euclidean space \mathbb^n; that is, the transformations of that space that preserve the Euclidean distance between any two points (also called Euclidean transformations). ...
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
The space of oriented lines is a double covering map, 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
In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...
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; the integral is then performed with the natural measure on the space of
geodesic
In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
s.
More general forms exist, such as the kinematic formula of Chern.
Applications
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
A bounding volume hierarchy (BVH) is a tree structure on a set of geometric objects. All geometric objects, that form the leaf nodes of the tree, are wrapped in bounding volumes. These nodes are then grouped as small sets and enclosed within la ...
.
*Given compact convex subset
, let
be a random line, and
be a random hyperplane, then
where
be the expected shadow area of
S (that is,
T is the orthogonal projection to a random hyperplane of
\R^n), then by integrating Crofton formula first over
dp, then over
d\varphi, we get
\frac
= \frac
= 2\sqrt\pi\fracIn 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
Body of constant brightness, bodies of constant brightness.
See also
*
Buffon's noodle
In geometric probability, the problem of Buffon's noodle is a variation on the well-known problem of Buffon's needle, named after Georges-Louis Leclerc, Comte de Buffon who lived in the 18th century. This approach to the problem was published by Jo ...
* The
Radon transform
In mathematics, the Radon transform is the integral transform which takes a function ''f'' defined on the plane to a function ''Rf'' defined on the (two-dimensional) space of lines in the plane, whose value at a particular line is equal to the l ...
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
*
Steinhaus longimeter
The Steinhaus longimeter, patented by the professor Hugo Steinhaus, is an instrument used to measure the lengths of curves on maps.
Description
It is a transparent sheet of three grids, turned against each other by 30 degrees, each consisting ...
References
*
*{{cite book , first=L. A. , last=Santalo , year=1953 , title=Introduction to Integral Geometry , pages=12–13, 54 , id={{LCC, QA641.S3
External links
Cauchy–Crofton formula page with demonstration applets
Alice, Bob, and the average shadow of a cube a visualization of Cauchy's surface area formula.
Integral geometry
Measure theory
Differential geometry>T(S), /math> be the expected shadow area of
S (that is,
T is the orthogonal projection to a random hyperplane of
\R^n), then by integrating Crofton formula first over
dp, then over
d\varphi, we get
\frac
= \frac
= 2\sqrt\pi\fracIn 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
Body of constant brightness, bodies of constant brightness.
See also
*
Buffon's noodle
In geometric probability, the problem of Buffon's noodle is a variation on the well-known problem of Buffon's needle, named after Georges-Louis Leclerc, Comte de Buffon who lived in the 18th century. This approach to the problem was published by Jo ...
* The
Radon transform
In mathematics, the Radon transform is the integral transform which takes a function ''f'' defined on the plane to a function ''Rf'' defined on the (two-dimensional) space of lines in the plane, whose value at a particular line is equal to the l ...
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
*
Steinhaus longimeter
The Steinhaus longimeter, patented by the professor Hugo Steinhaus, is an instrument used to measure the lengths of curves on maps.
Description
It is a transparent sheet of three grids, turned against each other by 30 degrees, each consisting ...
References
*
*{{cite book , first=L. A. , last=Santalo , year=1953 , title=Introduction to Integral Geometry , pages=12–13, 54 , id={{LCC, QA641.S3
External links
Cauchy–Crofton formula page with demonstration applets
Alice, Bob, and the average shadow of a cube a visualization of Cauchy's surface area formula.
Integral geometry
Measure theory
Differential geometry