In mathematics, integral geometry is the theory of measures on a geometrical space invariant under the

symmetry group In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the amb ...

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 transform In mathematics, an integral transform maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily characterized and manipulated than i ...

s such as 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 ...

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 Santaló and Wilhelm Blaschke was in this connection. It follows from the classic theorem of Crofton expressing the

length Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Inte ...

of a plane

curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...

as an

expectation Expectation or Expectations may refer to: Science * Expectation (epistemic) * Expected value, in mathematical probability theory * Expectation value (quantum mechanics) * Expectation–maximization algorithm, in statistics Music * ''Expectation' ...

of the number of intersections with a

random In common usage, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. Individual rando ...

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 In mathematics, the affine group or general affine group of any affine space over a field is the group of all invertible affine transformations from the space into itself. It is a Lie group if is the real or complex field or quaternions. ...

of the plane acts. A

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''. The difference between a probability measure and the more g ...

is sought on this space, invariant under the symmetry group. If, as in this case, we can find a unique such invariant measure, then 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 A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true premises, leads to a seemingly self-contradictory or a logically u ...

es—for example Bertrand's paradox.) We can therefore say that integral geometry in this sense is the application of

probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...

(as axiomatized by

Kolmogorov Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Sovi ...

) 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 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 transitively. The elements of ...

s of

Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addit ...

s; and the evaluation of integrals of 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 application ...

s. 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 processes 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 in the Euclidean setting. Subsequently Hadwiger-type theorems were established in various settings, notably in hermitian geometry, using advanced tools from valuation theory. The more recent meaning of integral geometry is that of Sigurdur Helgason and

Israel Gelfand Israel Moiseevich Gelfand, also written Israïl Moyseyovich Gel'fand, or Izrail M. Gelfand ( yi, ישראל געלפֿאַנד, russian: Изра́иль Моисе́евич Гельфа́нд, uk, Ізраїль Мойсейович Гел� ...

.I.M. Gel’fand (2003) ''Selected Topics in Integral Geometry'', American Mathematical Society It deals more specifically with integral transforms, modeled on the

. 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''.

Classical context

Notes

Further reading