- Integral geometry
In
mathematics , the term "integral geometry" is used in two ways, which, although related, imply different views of the content of the subject.Cases
The more traditional usage is that of Santalo and Blaschke. It follows from the classic theorem of Crofton expressing the
length of a planecurve as an expectation of the number of intersections with arandom 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. Aprobability 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 someparadox es.)We can therefore say that "integral geometry in the sense of Santalo", is the application of
probability theory (as axiomatized byKolmogorov ) in the context of theErlangen programme of Klein. The content of the theory is effectively that of invariant (smooth) measures on (preferablycompact )homogeneous space s ofLie group s; and the evaluation of integrals ofdifferential 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 variousstochastic process es concerned with geometric and incidence questions.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 theRadon 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".
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