Integral geometry
Encyclopedia
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
) transformations from the space of functions on one geometrical space to the space of functions on another geometrical space. Such transformations often take the form of integral transforms such as the Radon transform
and its generalizations.
and Wilhelm Blaschke
was in this connection. It follows from the classic theorem of Crofton
expressing the length
of a plane curve
as an expectation
of the number of intersections with a random line. Here the word 'random' must be interpreted as subject to correct symmetry considerations.
There is a sample space of lines, one on which the affine group
of the plane acts. A probability measure
is sought on this space, invariant under the symmetry group. If, as in this case, we can find a unique such invariant measure, that solves the problem of formulating accurately what 'random line' means; and expectations become integrals with respect to that measure. (Note for example that the phrase 'random chord of a circle' can be used to construct some paradox
es.)
We can therefore say that integral geometry in this sense is the application of probability theory
(as axiomatized by Kolmogorov) in the context of the Erlangen programme of Klein. The content of the theory is effectively that of invariant (smooth) measures on (preferably compact
) homogeneous space
s of Lie group
s; and the evaluation of integrals of differential form
s arising.
A very celebrated case is the problem of Buffon's needle
: drop a needle on a floor made of planks and calculate the probability the needle lies across a crack. Generalising, this theory is applied to various stochastic process
es concerned with geometric and incidence questions. See stochastic geometry
.
One of the most interesting theorems in this form of integral geometry is Hadwiger's theorem
.
The more recent meaning of integral geometry is that of Israel Gelfand
. It deals more specifically with integral transforms, modelled on the Radon transform
. Here the underlying geometrical incidence relation (points lying on lines, in Crofton's case) is seen in a freer light, as the site for an integral transform composed as pullback onto the incidence graph and then push forward.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, integral geometry is the theory of measures
Measure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...
on a geometrical space invariant under the symmetry group
Symmetry group
The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...
of that space. In more recent times, the meaning has been broadened to include a view of invariant (or equivariant
Equivariant
In mathematics, an equivariant map is a function between two sets that commutes with the action of a group. Specifically, let G be a group and let X and Y be two associated G-sets. A function f : X → Y is said to be equivariant iffor all g ∈ G and all x in X...
) transformations from the space of functions on one geometrical space to the space of functions on another geometrical space. Such transformations often take the form of integral transforms such as the Radon transform
Radon transform
thumb|right|Radon transform of the [[indicator function]] of two squares shown in the image below. Lighter regions indicate larger function values. Black indicates zero.thumb|right|Original function is equal to one on the white region and zero on the dark region....
and its generalizations.
Classical context
Integral geometry as such first emerged as an attempt to refine certain statements of geometric probability theory. The early work of Luis SantalóLuis Santaló
Luís Antoni Santaló Sors was a Spanish mathematician.He graduated from the University of Madrid and he studied at the University of Hamburg, where he received his Ph.D. in 1936. His advisor was Wilhelm Blaschke...
and Wilhelm Blaschke
Wilhelm Blaschke
Wilhelm Johann Eugen Blaschke was an Austro-Hungarian differential and integral geometer.His students included Shiing-Shen Chern, Luis Santaló, and Emanuel Sperner....
was in this connection. It follows from the classic theorem of Crofton
Crofton formula
In mathematics, the Crofton formula, named after Morgan Crofton , 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:...
expressing the length
Length
In geometric measurements, length most commonly refers to the longest dimension of an object.In certain contexts, the term "length" is reserved for a certain dimension of an object along which the length is measured. For example it is possible to cut a length of a wire which is shorter than wire...
of a plane curve
Curve
In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...
as an expectation
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
of the number of intersections with a random line. Here the word 'random' must be interpreted as subject to correct symmetry considerations.
There is a sample space of lines, one on which the affine group
Affine group
In mathematics, the affine group or general affine group of any affine space over a field K is the group of all invertible affine transformations from the space into itself.It is a Lie group if K is the real or complex field or quaternions....
of the plane acts. A probability measure
Probability measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as countable additivity...
is sought on this space, invariant under the symmetry group. If, as in this case, we can find a unique such invariant measure, that solves the problem of formulating accurately what 'random line' means; and expectations become integrals with respect to that measure. (Note for example that the phrase 'random chord of a circle' can be used to construct some paradox
Paradox
Similar to Circular reasoning, A paradox is a seemingly true statement or group of statements that lead to a contradiction or a situation which seems to defy logic or intuition...
es.)
We can therefore say that integral geometry in this sense is the application of probability theory
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
(as axiomatized by Kolmogorov) in the context of the Erlangen programme of Klein. The content of the theory is effectively that of invariant (smooth) measures on (preferably compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
) homogeneous space
Homogeneous space
In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts continuously by symmetry in a transitive way. A special case of this is when the topological group,...
s of Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
s; and the evaluation of integrals of differential form
Differential form
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...
s arising.
A very celebrated case is the problem of Buffon's needle
Buffon's needle
In mathematics, Buffon's needle problem is a question first posed in the 18th century by Georges-Louis Leclerc, Comte de Buffon:Buffon's needle was the earliest problem in geometric probability to be solved; it can be solved using integral geometry...
: drop a needle on a floor made of planks and calculate the probability the needle lies across a crack. Generalising, this theory is applied to various stochastic process
Stochastic process
In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...
es concerned with geometric and incidence questions. See stochastic geometry
Stochastic geometry
In mathematics, stochastic geometry is the study of random spatial patterns. At the heart of the subject lies the study of random point patterns...
.
One of the most interesting theorems in this form of integral geometry is Hadwiger's theorem
Hadwiger's theorem
In integral geometry , Hadwiger's theorem characterises the valuations on convex bodies in Rn. It was proved by Hugo Hadwiger.-Valuations:...
.
The more recent meaning of integral geometry is that of Israel Gelfand
Israel Gelfand
Israel Moiseevich Gelfand, also written Israïl Moyseyovich Gel'fand, or Izrail M. Gelfand was a Soviet mathematician who made major contributions to many branches of mathematics, including group theory, representation theory and functional analysis...
. It deals more specifically with integral transforms, modelled on the Radon transform
Radon transform
thumb|right|Radon transform of the [[indicator function]] of two squares shown in the image below. Lighter regions indicate larger function values. Black indicates zero.thumb|right|Original function is equal to one on the white region and zero on the dark region....
. Here the underlying geometrical incidence relation (points lying on lines, in Crofton's case) is seen in a freer light, as the site for an integral transform composed as pullback onto the incidence graph and then push forward.