Integral Geometry

a branch of mathematics that studies certain special numerical characteristics (“measures”) for sets of points, lines, planes, and other geometrical objects, calculated, as a rule, with the aid of integration. Here the “measure” must satisfy the following two requirements: (1) additivity (the measure of a set consisting of several parts is equal to the sum of the measures of these parts) and (2) invariance with respect to motion (two sets differing only in position have the same measure). Integral geometry primarily includes problems of finding lengths, areas, and volumes, which are solved by means of integration (involving simple, double, and triple integrals, respectively).

The impetus for the development of integral geometry was provided by problems related to so-called geometric probabilities, defined as the ratio of the measure of the set of favorable cases to the measure of the set of all possible cases (by analogy with the classical definition of probability as the ratio of the number of favorable cases to the number of all possible cases). The first and most famous example is Button’s needle problem (1777). Consider a plane covered by a family of parallel lines such that any two adjacent lines are a fixed distance h apart. A thin cylindrical needle of length l < h drops to the plane in a random manner. What is the probability that the needle will intersect one of the lines? This problem is equivalent to the following: What is the probability that a randomly chosen line that intersects a circle (of diameter h) intersects a given segment of length l < h whose midpoint is at the center of the circle? This probability is defined as the ratio of the “measure” of the set of lines intersecting the given interval to the measure of the set of lines intersecting the given circle. The measure of a set of lines that intersect a convex figure of finite perimeter is introduced in such a way as to satisfy the two requirements formulated above: additivity and invariance under motion.

In the case of the set of all lines intersecting a line segment, the measure of this set must, by virtue of invariance with respect to motion, be a function only of the length of the segment. The requirement of additivity of measure implies that this function f(x) must be additive: f(x + y) = f(x) + f(y). But then f(x) = Cx, where C is a constant. Thus, in the plane, the measure of the set of all lines intersecting a given segment must be proportional to its length. The proportionality coefficient is conveniently taken to be 2, that is, it is agreed that the number 2 is taken to be the measure of the set of lines that intersect a segment of length 1. Then the measure of the set of lines intersecting any segment turns out to be double its length.

It can be shown that the measure of the set of lines each of which intersects the boundary of a given convex polygon (in two points) is equal to its perimeter.

Proceeding, finally, to the set of lines intersecting a closed convex curve (an “oval”), it is not difficult to establish the fact that in a plane the measure of the set of lines intersecting the given convex curve must be equal to the length of this curve.

In Buffon’s needle problem the measure of the set of favorable cases is double the length (21) of the needle, and the measure of the set of possible cases is the circumference (πh) of a circle of diameter h; therefore the desired probability p = 2l/πh. This result has been experimentally tested many times. In one such experiment, 5,000 throws were performed; with l = 36 mm and h = 45 mm, a frequency of intersection of 0.5064 was obtained, which gives an approximate value of π equal to 3.1596.

With certain modifications, the above theory may be carried over to sets of lines that intersect nonconvex figures. In general, for two-parameter sets of lines in a plane, a measure (μ) can be defined by the formula μ = ∫∫ dρdϕ, where ρ and ϕ are the polar coordinates of the projection of the origin to a line. If the line is defined by the equation ux + vy = 1 (x and y are the rectangular coordinates of a point on the line), then

By the turn of the 20th century the investigations of integral geometry were still connected with geometric probabilities (works of the British mathematician M. Crofton and the French mathematician H. Poincaré), but already in the work of the French mathematician E. Cartan (1896) these studies became part of a general theory of integral invariants, and in the 1920’s they formed an independent theory with various applications: in geometry “in the large,” primarily in the study of convex regions, and in geometric optics and the theory of radiation.