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The more traditional usage is that of Santalo and Blaschke . 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 the sense of Santalo, 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 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 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''. |
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