Determination of the Moments of a Random Chord in an Ellipsoid
Abstract Generated abstract
This paper applies a general method for finding statistical properties of random chords in convex bodies to the case of an ellipsoid with arbitrary orientation. Using coordinate transformations, projection onto a plane, and integration over the projected region, it derives formulas for the mean chord length, second moment of chord length, projection area, and related orientation-dependent quantities in terms of the ellipsoid semiaxes and direction cosines or Euler angles. The results show, in particular, that the product of the mean chord length in a fixed direction and the corresponding projection area equals the ellipsoid volume, and they provide integral expressions for averaging over all orientations. Several special cases are evaluated, including the sphere and rotations about principal axes, with some results expressed through complete elliptic integrals.
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DETERMINATION OF THE MOMENTS OF A RANDOM CHORD IN AN ELLIPSOID
I. E. NAATS
(Presented by the scientific seminar of the Department of Higher Mathematics, TPI)
In the present paper we consider the application of methods for determining the moments of a random chord in a convex body, described in paper [1] of the present collection, using the ellipsoid as an example.
Let the position of the ellipsoid in space be defined in the system \(S\), with \(S'\) the coordinate system associated with the principal axes of the ellipsoid. The rotation of the ellipsoid about the point \(O\) (Fig. 1) may be specified by the rotation of the system \(S'\) about the point \(O\). The corresponding coordinate transformation is written in the form
Fig. 1
\[ \left. \begin{aligned} x'&=l_1x+m_1y+n_1z\\ y'&=l_2x+m_2y+n_2z\\ z'&=l_3x+m_3y+n_3z \end{aligned} \right\}. \tag{1} \]
Then the equation of the surface of the ellipsoid in the system \(S\) takes the form
\[ \frac{(l_1x+m_1y+n_1z)^2}{a^2} + \frac{(l_2x+m_2y+n_2z)^2}{b^2} + \frac{(l_3x+m_3y+n_3z)^2}{c^2} =1. \tag{2} \]
We shall project the ellipsoid onto some plane \(W\) in the direction of the \(x\)-axis of the system \(S\). The length of a chord of the ellipsoid in the direction of the \(x\)-axis is determined as the difference of the roots \(x_1\) and \(x_2\) of equation (2), solved with respect to \(x\), i.e.,
\[ l_x=x_1-x_2 . \tag{3} \]
Accordingly, we have the equation of the boundary of the projection of the ellipsoid \(P_x\) onto the plane \(W\) in the direction of \(x\):
\[ l_x=0 . \tag{4} \]
The mean value of the chord of the ellipsoid in the direction of the \(x\)-axis for some fixed position of the system \(S'\) (also an ellipsoid) will be determined by the integral
\[ \overline{l}_x=\frac{1}{P_x}\int_{P_x} l_x\,dP . \tag{5} \]
Accordingly, the second moment is
\[ \overline{l_x^2}=\frac{1}{P_x}\int_{P_x} l_x^2\,dP . \tag{6} \]
Omitting the calculations, we give the final result:
\[ \overline{l}_x=\frac{4}{3}\, \frac{abc}{\sqrt{b^2c^2l_1^2+a^2c^2l_2^2+a^2b^2l_3^2}}, \tag{7} \]
\[ P_x=\pi\sqrt{b^2c^2l_1^2+a^2c^2l_2^2+a^2b^2l_3^2}, \tag{8} \]
\[ \overline{l_x^2}= \frac{2a^2b^2c^2}{b^2c^2l_1^2+a^2c^2l_2^2+a^2b^2l_3^2}. \tag{9} \]
From relations (7) and (8), in particular, it is clear that the quantity
\[ \overline{l}_x\cdot P_x=\mathrm{const}=\frac{4}{3}\pi abc, \]
i.e., the volume of the ellipsoid.
Introduce the notation
\[ L(l_1,l_2,l_3)=b^2c^2l_1^2+a^2c^2l_2^2+a^2b^2l_3^2 \tag{10} \]
and pass from the direction cosines \(l_1,l_2,l_3\) to the system of Euler angles \(\vartheta,\psi,\varphi\). As a result we obtain
\[ \begin{aligned} L(\vartheta,\psi,\varphi)=\;& 2c^2(a^2-b^2)\cos\vartheta\cdot\cos\varphi\cdot\cos\psi\cdot\sin\psi\cdot\sin\varphi \\ &+c^2\cos^2\vartheta\,\sin^2\psi\,(a^2\cos^2\varphi+b^2\sin^2\varphi) \\ &+c^2\cos^2\psi\,(b^2\cos^2\varphi+a^2\sin^2\varphi) +a^2b^2\sin^2\vartheta\cdot\sin^2\psi . \end{aligned} \tag{11} \]
Taking (11) into account, we have:
\[ \overline{l}_x=\frac{4}{3}\, \frac{abc}{\sqrt{L(\vartheta,\psi,\varphi)}} , \tag{12} \]
\[ P_x=\pi \sqrt{L(\vartheta,\psi,\varphi)}, \tag{13} \]
\[ \bar l_x^{\,2}=\frac{2a^2b^2c^2}{L(\vartheta,\psi,\varphi)}. \tag{14} \]
To determine the mean values of the indicated quantities over all possible orientations of the ellipsoid with respect to the plane \(W\), it is necessary to compute the integrals with respect to \(\vartheta,\psi,\varphi\) of the corresponding expressions. We have
\[ \bar l=\frac{32}{3\pi^3}abc \int_0^{\pi/2} d\vartheta \int_0^{\pi/2} d\psi \int_0^{\pi/2} \frac{d\varphi}{\sqrt{L(\vartheta,\psi,\varphi)}}, \tag{15} \]
\[ \overline{l^2}=\frac{16}{\pi^3}a^2b^2c^2 \int_0^{\pi/2} d\vartheta \int_0^{\pi/2} d\psi \int_0^{\pi/2} \frac{d\varphi}{L(\vartheta,\psi,\varphi)}, \tag{16} \]
\[ \bar P=\frac{8}{\pi^2} \int_0^{\pi/2} d\vartheta \int_0^{\pi/2} d\psi \int_0^{\pi/2} \sqrt{L(\vartheta,\psi,\varphi)}\,d\varphi, \tag{17} \]
\[ \overline{P^2}=\frac{8}{\pi^2} \int_0^{\pi/2} d\vartheta \int_0^{\pi/2} d\psi \int_0^{\pi/2} L(\vartheta,\psi,\varphi)\,d\varphi. \tag{18} \]
From the general expressions (12—14), as well as (15—16), one can consider certain special cases.
- Let \(a=b=c=R\), i.e., the ellipsoid becomes a sphere; then
\[ L(\vartheta,\psi,\varphi)=R^4 \]
and, accordingly,
\[ \bar l=\frac{4}{3}R, \tag{19} \]
\[ \overline{l^2}=2R^2. \tag{20} \]
- Let us determine the mean value of the chord of an ellipsoid due to rotation of the latter about the \(y\)-axis. In this case
\[ -\pi<\vartheta<\pi,\qquad \varphi=\psi=\pi/2 \quad\text{and} \]
\[ L(\vartheta)=b^2(c^2\cos^2\vartheta+a^2\sin^2\vartheta). \tag{21} \]
Accordingly, we have
\[ \bar l=\frac{8}{3\pi}ac \int_0^{\pi/2} \frac{d\vartheta}{\sqrt{c^2\cos^2\vartheta+a^2\sin^2\vartheta}} = \frac{8a}{3\pi}\,F(k), \tag{22} \]
where \(F(k)\) is the complete elliptic integral of the second kind,
\[ k^2=(a^2-c^2)/a^2\qquad (a>c). \]
For the remaining quantities we have
\[ \overline{l^2}=2ac, \tag{23} \]
\[ \bar P=2ab\int_0^{\pi/2}\sqrt{1-k^2\sin^2\vartheta}\,d\vartheta =2ab\,E(k), \tag{24} \]
where \(E(k)\) is the complete elliptic integral of the first kind, and
\[ \overline{P}^{\,2}=\frac{\pi^2 b^2}{2}(a^2+c^2). \tag{25} \]
- When the ellipsoid is rotated about the \(z\)-axis, we have \(\vartheta=0\), \(\psi=0\), \(0\leq \varphi \leq 2\pi\), and
\[ L(\varphi)=c^2(b^2\cos^2\varphi+a^2\sin^2\varphi). \tag{26} \]
The values of the remaining quantities are analogous to the results (22–25).
- When the ellipsoid is rotated about the \(x\)-axis, we have
\[ \varphi=\psi=0,\qquad 0\leq \vartheta\leq \pi \quad \text{and} \]
\[ L(\vartheta,\psi,\vartheta)=c^2b^2, \]
\[ \overline{l}=\frac{4}{3}a, \tag{27} \]
\[ \overline{l^2}=2a^2. \tag{28} \]
The same result is valid when the ellipsoid is rotated about any axis, if the direction of projection coincides with the direction of this axis. Thus, the mean values of the random chords of the ellipsoid \(\overline{l}_x\), \(\overline{l}_y\), \(\overline{l}_z\) in three mutually perpendicular directions coinciding with its principal axes are equal, respectively, to
\[ \frac{4}{3}a,\ \frac{4}{3}b \ \text{and}\ \frac{4}{3}c. \]
Expressions (12), (13), and (14) may be used as the basis for numerical computation of the values corresponding to the quantity for given \(a\), \(b\), and \(c\).
References
- I. E. Naats. On the determination of the statistical properties of a random chord in a convex body. (This collection).