On the Determination of the Statistical Properties of a Random Chord in a Convex Body
Abstract Generated abstract
This paper examines how to determine moments of the length of a random chord in a convex body, a problem arising in probabilistic geometric models of random media based on random secants. It formulates the invariant measure on the space of lines, derives the standard relation between mean chord length, volume, and surface area, and uses projection identities to obtain practical bounds when direct integration is difficult. The work gives estimates for elementary convex bodies such as ellipsoids, cubes, and parallelepipeds, including two-sided bounds based on projection areas and comparison with the sphere of equal volume. Numerical examples indicate how the sharpness of the estimates depends on body shape, and the paper notes that simulation may be needed for more accurate higher moments.
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ON THE DETERMINATION OF STATISTICAL PROPERTIES OF A RANDOM CHORD IN A CONVEX BODY
I. E. NAATS
(Presented by the scientific seminar of the Department of Higher Mathematics, TPI)
In studying the statistics of a random medium using a probabilistic-geometric model of the filling of three-dimensional space by a system of convex bodies by the method of a random secant, it becomes necessary to estimate the moments of a random chord in the bodies of the filling [1]. For practical purposes it is usually possible to restrict oneself to the first two moments.
In the present work we consider questions of the method for calculating the moments of a random chord in a convex body, and estimates of the first moment are given for certain bodies of elementary form. Let there be given some convex body \(T\) and a set of random chords \(l\) in this body. The set of random chords, in turn, may be regarded as generated by the set of all lines \(J\) in three-dimensional space \(R_3\) having at least one common point with the body \(T\). In turn, the set \(J\) is a subset of the set of all lines in \(R_3\).
A line in \(R_3\) can be specified by a system of two equations
\[ \begin{aligned} x&=p_1z+q_1,\\ y&=p_2z+q_2 \end{aligned} \tag{1} \]
The parameters \(p_1, p_2, q_1\), and \(q_2\) together form the parametric space \(P\) of the set of lines. The parametric space \(P\) uniquely determines the set of lines in \(R_3\).
An element of measure of the space \(P\), invariant with respect to the group of rotations and translations in \(R_3\), is determined by the expression [1]:
\[ dP=(1+p_1^2+p_2^2)^{-2}dp_1dp_2dq_1dq_2. \tag{2} \]
To calculate the values of a random chord it is necessary to determine the measure of the set \(P'\), which is a subset of \(P\) and corresponds to the set of lines \(l\) intersecting the given body \(T\). Formally this can be done by solving the system
\[ \begin{aligned} F(x,y,z)&=0\\ x&=p_1z+q_1,\\ y&=p_2z+q_2 \end{aligned} \tag{3} \]
where \(F(x,y,z)=0\) is the equation of the surface of the body \(T\), and by investigating the region inside which the roots of the system \(z_1(x_1,y_1)\) and \(z_2(x_2,y_2)\) are real and distinct. Equality of the roots to each other corresponds to the set of tangents to the body \(T\). The length of the chord is determined as the distance between the roots in \(R_3\). As a result, the mean value of the chord is determined by the expression
\[ \bar l=\frac{1}{m(P')}\int_{m(P')} l\,dP', \tag{4} \]
where \(m(P')\) is the measure of the parametric space \(P'\) corresponding to the set of random secants. The remaining moments of the quantity \(l\) are determined analogously. The magnitude of the measure of the set \(P'\) is determined by the topological properties of the body \(T\). In particular, it has been proved that for a convex body with surface \(S\) [1]
\[ m(P')=\frac{1}{2}\pi S. \tag{5} \]
It is very difficult to use the described method in its general form for practical calculations. Therefore one usually resorts to the following device. First one constructs the subset of secants corresponding to some chosen direction \(\vec n\). The measure of this subset, as is easy to determine, is equal to
\[ dF=\frac{1}{2}F(\vec n)d\omega, \tag{6} \]
where \(F(\vec n)\) is the area of the projection of the body \(T\) onto some plane perpendicular to the direction \(\vec n\), and \(d\omega\) is an element of solid angle. Then the total measure of the set of secants is determined by the expression
\[ m(P')=\int dF=\frac{1}{2}\int F(\vec n)d\omega=2\pi \bar F. \tag{7} \]
The mean value of the chord is determined by the integral
\[ \bar l=\frac{1}{m(P')}\cdot\frac{1}{2}\iint l\,dF(\vec n)d\omega. \tag{8} \]
It is not difficult to see that
\[ \int l\,dF(\vec n)=V \tag{9} \]
and
\[ \bar l(\vec n)\cdot F(\vec n)=V, \tag{10} \]
where \(V\) is the volume of the body and \(\bar l(\vec n)\) is the mean value of the random chord on the set of secants parallel to the given direction \(\vec n\). Hence, taking (5) into account, two important relations can be written:
\[ \bar l=\frac{4V}{S}, \tag{11} \]
\[ S=4\bar F, \tag{12} \]
where \(\bar F\) is the mean projection of the body onto a plane in the direction \(\vec n\) when this body is rotated relative to this plane.
The calculation of the mean value of a random chord and of its higher-order moments is very complicated and laborious even for bodies of elementary shape. However, in a number of cases one can get by with estimates of the indicated quantities. In particular, using relations (10), (11), (12), one can construct for the mean value \(\bar l\) a number of estimates for bodies of elementary shape.
Let a convex body of volume \(V\) rotate relative to the vector \(\vec n\) and to the projection plane perpendicular to \(\vec n\). Then the quantities \(\bar l(\vec n)\) and \(F(\vec n)\) will vary depending on the orientation of the body in space. However, the product of these quantities remains constant and equal to the volume \(V\), as follows from (10). Therefore, if it is possible for the given body to determine the largest and smallest values of the orthogonal projection onto some plane (i.e., \(F_{\max}\) and \(F_{\min}\)), then for \(\bar l\) the inequality
\[ \frac{V}{F_{\max}}<\bar l<\frac{V}{F_{\min}} . \tag{13} \]
holds.
In relation (13), equality is attained for a sphere, for which \(F(\vec n)=\mathrm{const}\) for all positions in space relative to the projection plane.
As an example one may consider an ellipsoid, for which \(F_{\max}=\pi ab\) and \(F_{\min}=\pi bc\), with semiaxes satisfying the condition \(a>b>c\). For such an ellipsoid, from (13) we have the inequality
\[ \frac{4}{3}c<\bar l<\frac{4}{3}a . \tag{14} \]
For a cube with edge \(a\),
\[ F_{\min}=a^2,\quad F_{\max}=a^2\left(1+\frac{1}{\sqrt{2}}\right) \]
and, accordingly,
\[ \frac{1}{1+\frac{1}{\sqrt{2}}}\,a<\bar l<a . \tag{15} \]
For a convex body of arbitrary shape one can construct a lower estimate for \(\bar l\), using the following fact: for a convex body, the area of the projection \(F\) in any direction satisfies the inequality [2]
\[ F^2<F_1^2+F_2^2+F_3^2, \tag{16} \]
where \(F_1, F_2, F_3\) are, respectively, the values of the projections of this body in three mutually perpendicular directions. Hence it is not difficult to obtain a lower estimate for \(\bar l\), namely:
\[ \frac{V}{\sqrt{F_1^{\,2}+F_2^{\,2}+F_3^{\,2}}}=l' \leq \overline{l}. \tag{17} \]
Equality in (17) is attained for a plane figure; therefore estimate (17) is the more accurate, the more elongated the body is with respect to one of the axes or directions.
For a convex body of volume \(V\), an upper estimate for the mean value of a random chord can be constructed by using the remarkable parametric property of the sphere, which consists in the fact that among all convex bodies of a given volume the sphere has the least surface area [2]. Then from (11) there follows the upper estimate for the mean value of the random chord of a convex body
\[ \overline{l}\leq \overline{l}_0, \tag{18} \]
where \(\overline{l}_0\) is the mean value of a random chord in a sphere of volume \(V\). Combining (17) and (18), we obtain a two-sided estimate for the value \(\overline{l}\).
Table 1
| Value of the quantities \(a, b, c\) (for an ellipsoid, the semiaxes; for a parallelepiped, the sides) | Value of the quantity \(l'\) (for an ellipsoid and for a parallelepiped) | Value of the quantity \(l_0\) (for an ellipsoid and a parallelepiped) | Mean value of the quantity \(\overline{l}\) for a parallelepiped |
|---|---|---|---|
| \(a=1,\ b=1,\ c=1\) | 0,7737 1,106 |
1,3333 1,6349 |
1,3333 |
| 1; 0,75; 0,75 | 0,6247 0,9370 |
1,1006 1,3661 |
1,0908 |
| 1; 0,75; 0,25 | 0,5121 0,7682 |
0,9655 1,1933 |
0,9230 |
| 1; 0,75; 0,5 | 0,3078 0,4615 |
0,7631 0,9472 |
0,6316 |
| 1; 0,5; 0,5 | 0,4444 0,6666 |
0,8399 1,0342 |
0,8000 |
| 1; 0,5; 0,25 | 0,2909 0,4364 |
0,6666 0,8276 |
0,5714 |
| 1; 0,25; 0,25 | 0,2321 0,3481 |
0,5291 0,6567 |
0,4444 |
\[ l' < \overline{l} < \overline{l}_0. \tag{19} \]
Let us give the value of the quantity \(\overline{l}_0\) corresponding to bodies of elementary form. In particular, for an ellipsoid with semiaxes \(a, b, c\) we have
\[ \overline{l}_0=\frac{4}{3}\sqrt[3]{abc}, \tag{20} \]
for a cube with edge \(a\)
\[ \overline{l}_0=\left(\frac{4}{3}\right)^{-\frac{2}{3}}\pi^{-\frac{1}{3}}a. \tag{21} \]
and for a parallelepiped with sides \(a, b, c\)
\[ \bar l_0=\left(\frac{4}{3}\right)^{-\frac{2}{3}}\pi^{-\frac{1}{3}}\sqrt[3]{abc}. \tag{22} \]
The efficiency of the estimates obtained may be judged from the numerical examples in Table 1.
From the example (Table 1) of a parallelepiped, for which \(\bar l\) is determined quite simply from (11), it is seen that the true value of \(\bar l\), when the ratio of the dimensions of the body is varied, may be closer to \(l'\) or to \(\bar l_0\). If the body is more elongated, then \(\bar l\) is closer to \(l'\); if all the dimensions of the body are closer to one another, \(\bar l\) is closer to \(\bar l_0\). Taking into account that the computation of the values of \(\bar l\) for a convex body is very laborious and complicated—this applies especially to moments of higher order—the estimates presented may be regarded as quite satisfactory for practical purposes. This is especially true of the quantity \(\bar l_0\), since in the study of probabilistic-geometric models of random media one usually has to resort to the model of filling space with balls [1]. If more accurate estimates of the values of \(\bar l\) and its moments are required, they can be obtained by digital simulation on an electronic computer of the geometric model of the intersection of a random line with a convex body. In these problems, the method of statistical trials is evidently the only universal method.
REFERENCES
- M. G. Kendall, P. Moran. Geometrical Probability. London, 1963.
- V. Blaschke. Circle and Sphere. “Nauka,” Moscow, 1967.