Reports of the Academy of Sciences of the USSR
1969. Volume 187, No. 3

UDC 519.212.3

MATHEMATICS

R. V. Ambartsumian

INTERSECTIONS OF COMPLEX CURVES BY RANDOM STRAIGHT LINES

(Presented by Academician Yu. V. Linnik, 27 XII 1968)

The classical Crofton theorem on the number of intersections of a piecewise-smooth plane curve \(\mathscr L\) with straight lines states that

\[ \int n(g)\,\dot g = 2L . \tag{1} \]

Here \(\dot g\) is the (normalized) invariant density of lines on the plane with respect to motions; \(n(g)\) is the number of intersections of the line \(g\) with the curve \(\mathscr L\), and \(L\) is the length of the curve \(\mathscr L\) (see \((^1)\)).

Equality (1) is given the form of a probabilistic statement when only lines intersecting some convex region \(D\) containing \(\mathscr L\) entirely are considered. Indeed, introducing the function of lines

\[ \delta_D(g)= \begin{cases} 1, & \text{if } g \text{ intersects } D,\\ 0, & \text{otherwise,} \end{cases} \]

on the basis of (1) one may write

\[ \int n(g)\,\frac{\dot g\delta_D(g)}{U}=2\frac{L}{U}. \tag{2} \]

Here and below \(U\) is the length of the perimeter of \(D\), and the integral on the left is understood as the mean number of intersections of \(\mathscr L\) with a random line having probability distribution with density \(\dot g\delta_D(g)U^{-1}\). In what follows, a random line with such a distribution will be denoted by \(\mathscr M\).

Although results (1) and (2) are 100 years old, little has been known up to now about the distribution of the number of intersections of the random line \(\mathscr M\) with \(\mathscr L\). Almost the only results in this direction are those of Sylvester \((^2)\) for the case when \(\mathscr L\) is a union of three nonintersecting convex contours, i.e., when a direct reduction of the problem to Crofton’s result on mean numbers is possible, and of Zuluaga \((^6)\), who considered the relation between the topological properties of curves and the supports of the distribution of the number of their intersections with \(\mathscr M\).

The method of invariant imbedding, first applied in problems of geometry in \((^3,^4)\), leads to the determination of the distribution of the number of intersections of \(\mathscr L\) with \(\mathscr M\) under the most general assumptions concerning \(\mathscr L\). In saying this, we have in mind the reduction of integration over a two-dimensional manifold to integration over one- and zero-dimensional manifolds. In particular, equalities (1) and (2) have the same meaning.

The method of invariant imbedding itself is directly applicable only when \(\mathscr L\) is taken to be a polygonal line, and consists in the fact that the problem for the random line \(\mathscr M\) is considered as a limit problem for an analogous problem posed for a suitable random circle, when the radius of the circle tends to \(\infty\).

Let us make this statement precise. By definition, a random circle \(C(r)\) has constant radius \(r\), and its center is distributed uniformly inside the “fundamental disk” of radius \(R\) with center at the origin. \(R\) is chosen so large that the distance from \(\mathscr L\) to the boundary of the fundamental disk exceeds \(r\).

Let \(p_k(r)\) denote the probability of \(k\) intersections of \(C(r)\) with \(\mathscr L\); \(p_D(r)\), the probability of intersection of \(C(r)\) with the convex domain \(D\); \(p_k\), the probability of \(k\) intersections of \(\mathscr M\) with \(\mathscr L\).

Lemma. Suppose that \(\mathscr L\) is a broken line such that no three of its vertices lie on one straight line, and \(D\) is a convex polygon containing \(\mathscr L\). Then the limit of the ratio \(p_k(r)/p_D(r)\) exists (this ratio does not depend on \(R\)) as \(r\to\infty\), \(k>0\), and

\[ p_k=\lim_{r\to\infty}\frac{p_k(r)}{p_D(r)} \quad \text{for } k>0. \]

The derivatives \(\dfrac{d}{dr}p_k(r)\) and \(\dfrac{d}{dr}p_D(r)\) exist, and

\[ p_k=\lim_{r\to\infty}\frac{\dfrac{d}{dr}p_k(r)}{\dfrac{d}{dr}p_D(r)} \quad \text{for } k>0. \tag{3} \]

Using representation (3) to compute \(p_k\) for broken lines leads to the goal without special difficulties. In this, finding the asymptotics of the derivatives appearing in (3) is based on a number of simple geometric considerations, such as, for example, the fact that more than one vertex of the broken line \(\mathscr L\) falls into the annulus bounded by the circle \(C(r+h)\) and the concentric circle of radius \(r\) with probability of order of smallness \(o(h)\).

After computing the values \(p_k\) for broken lines, it is easy to carry out formally, and then justify, the limiting transition to piecewise-smooth curves. The result has an especially simple form in the case when \(\mathscr L\) is a closed, sufficiently smooth curve of finite length.

We additionally require that:

a) on \(\mathscr L\) there be no three distinct points at which the tangents to \(\mathscr L\) coincide (absence of triple tangents);

b) the curve \(\mathscr L\) have only a finite number of inflection points;

c) the curve \(\mathscr L\) have only a finite number of straight lines tangent to \(\mathscr L\) at two distinct points (double tangents);

d) the number of intersections of \(\mathscr L\) with a straight line be bounded.

It is not excluded that the curve \(\mathscr L\) consists of a finite number of closed components (as curves). We formulate the result for such curves. To this end we introduce three new random straight lines: \(T\), \(W^+\), \(W^-\). The random straight line \(T\), by definition, is a random tangent to \(\mathscr L\), and the point of tangency of \(T\) with \(\mathscr L\) is distributed uniformly along \(\mathscr L\).

By \(R_i\) (respectively, \(Q_i\)) we denote all those double tangents for which the curve \(\mathscr L\), locally, near the points of tangency, lies on one side (respectively, on different sides) of the double tangent itself. By \(r_i\) (respectively, \(q_i\)) we denote the length of the segment between the points of tangency of \(R_i\) (respectively, \(Q_i\)) with \(\mathscr L\).

The random straight line \(W^+\), by definition, takes the position \(R_i\) with probability equal to \(r_i/r\), \(r=\sum r_i\).

The random straight line \(W^-\), by definition, takes the position \(Q_i\) with probability equal to \(q_i/q\), \(q=\sum q_i\).

Let \(t_k\) denote the probability of \(k\) intersections of the random straight line \(T\) with \(\mathscr L\) (not counting the point of tangency); \(w_k^+\), the probability of \(k\) intersections of the random straight line \(W^+\) with \(\mathscr L\) (not counting the points of tangency); \(w_k^-\), the probability of \(k\) intersections of the random straight line \(W^-\) with \(\mathscr L\) (not counting the points of tangency).

Theorem*. For every smooth closed curve \(\mathscr L\) of finite length satisfying conditions a), b), c), and d), the relation

\[ p_k=\frac{r}{U}\left[2w_{k-2}^+ - w_k^+ - w_{k-4}^+\right] -\frac{q}{U}\left[2w_{k-2}^- - w_k^- - w_{k-4}^-\right]+ \]

* The corresponding results for more general (piecewise-smooth) curves can be obtained by approximating them by curves satisfying the conditions of the theorem.

\[ +\frac{L}{U}[t_{k-2}-t_k],\qquad k>0; \tag{4} \]

\[ p_0=1-\frac{r}{U}w_0^+ + \frac{q}{U}w_0^- - \frac{L}{U}t_0 . \]

Probabilities with negative indices should be taken to be equal to zero.

Let us note that a closed curve intersects any line in an even number of points (see \({}^{6}\)). If \(2N\) is the maximal number of intersections that can occur when \(\mathcal L\) is intersected by a line (i.e., \(p_{2N}>0\), but \(p_{2n}=0\) if \(n>N\)), then it is obvious that \(t_{2n}=0\) for \(n>N-1\) and \(w_{2n}^+=w_{2n}^-=0\) for \(n>N-2\). Taking this into account, from (4) we easily find the mathematical expectation of the random number \(\xi\) of intersections of \(\mathcal M\) with \(\mathcal L\):

\[ \bar n=E\xi=\sum kp_k=2L/U, \]

which coincides with (2). Further, putting

\[ \bar m=\sum kt_k, \]

we find the second moment of the distribution \(p_k\):

\[ E\xi^2=\frac{8(q-r)}{U}+4\frac{L}{U}(\bar m+1). \]

Thus the variance \(D\xi\), important for computational applications, is equal to

\[ D\xi=\frac{8(q-r)}{U}+4\frac{L}{U}(\bar m+1)-4\frac{L^2}{U^2}, \]

and the coefficient of variation, consequently, has the form

\[ D\frac{\xi}{\bar n}=\frac{2(q-r)U}{L^2}+\frac{U}{L}(\bar m+1)-1. \tag{5} \]

When estimating the magnitude of the coefficient of variation for a given \(\mathcal L\), it is important in practice to be able to neglect the first term in its expression (5). We confine ourselves to the remark that if \(\mathcal L\) is the union of some number of circles, then \(q-r<0\), i.e., the estimate

\[ D\frac{\xi}{\bar n}<\frac{U}{L}(\bar m+1)-1 \]

holds.

It is of interest to indicate a broader class of curves for which this estimate remains valid.

For the complete proofs, omitted in the present note, see \({}^{5}\).

Institute of Mathematics and Mechanics
Academy of Sciences of the ArmSSR

Received
20 XII 1968

CITED LITERATURE

\({}^{1}\) W. Blaschke, Vorlesungen über Integralgeometrie, N. Y., 1949. \({}^{2}\) J. J. Sylvester, Acta Math., 14, 185 (1891). \({}^{3}\) R. V. Ambartsumian, Probability Distributions in Geometric Combinatorics, Preprint, Institute of Mathematics and Mechanics, Academy of Sciences of the ArmSSR, Yerevan, 1968. \({}^{4}\) R. V. Ambartzumian, Studia Sci. Math. Hung., in press. \({}^{5}\) R. V. Ambartsumian, Intersections of a Plane Curve by Random Secants and Tangents, Preprint, VINITI, 1968. \({}^{6}\) R. Sulanke, Acta Math. Acad. Sci. Hung., 17 (3–4), 233 (1966).