Construction on an Analog Computer of Lines Lying on \(n\)-Dimensional Surfaces

I. G. Vintizenko, V. A. Tartakovsky

(Presented by the scientific seminar of the computing laboratory)

Constructions of spatial curves on analog computing machines are considered. For this purpose it is proposed to use a surface containing the required curve, and additional conditions characterizing it; the method of implicit functions plays the principal role here. Such constructions may serve for reproducing prescribed functions on an analog computer, obtaining graphical information, computing curvilinear integrals, etc.

Fig. 1. Scheme for obtaining equipotentials and computing the area of a section

Let an \(n\)-dimensional surface \(F(x_1 \ldots x_n)\) and a point \(M_0(x_1^0 x_2^0 \ldots x_n^0)\) be given; we express the rectangular Cartesian coordinates of the current point on the surface \(F(x_1 x_2 \ldots x_n)\) as follows:

\[ x_1 = x_1^0 + r \cos \varphi_1, \]
\[ x_2 = x_2^0 + r \cos \varphi_2, \]
\[ \ldots \ldots \ldots \]
\[ x_n = x_n^0 + r \cos \varphi_n. \]

Here \(r\) is the modulus of the \(n\)-dimensional radius vector with origin at the point \(M_0\), and \(\varphi\) are the angles between \(\vec r\) and the coordinate axes. If \(F(x_1 x_2 \ldots x_n)\) is closed and the point \(M_0\) is inside it, then \(r\) is always finite. If, however, \(F(x_1 x_2 \ldots x_n)\) is an open surface, then it is necessary to impose restrictions on the radius vector: \(|\vec r| \le R\)—an \(n\)-dimensional ball. Thus, in any cas-

in which case we obtain a surface closed around the point \(M_0\), and for any prescribed \(\varphi_i\) we shall have a point either on the surface \(F(x_1x_2\ldots x_n)\), or on the bounding surface \(|\vec r|=R\). They can be distinguished by the tracking error, which is equal to zero when the point lies on \(F(x_1x_2\ldots x_n)\), and is not equal to zero when \(|\vec r|=R\).

What is important is the continuity of the correspondence between the set of points of the curve and the set of values of the angles \(\varphi_i\). If a portion of the curve is not “visible” from the point \(M_0(x_1^0x_2^0\ldots x_n^0)\), then the functions \(z_i(t)\) undergo a discontinuity and this portion will be skipped. To avoid this, \(M_0\) is made to move along a preselected \(f(\varphi_1\varphi_2\ldots\varphi_n)\).

Computation of the Area of a Plane Figure

Since the function \(r=r(\varphi)\) is known, putting \(\varphi=t\), we obtain:

\[ S=\frac{1}{2}\int_{t_0}^{T} r^2(t)\,dt. \tag{1} \]

Computation of the Volume of a Three-Dimensional Body \(V(x_1x_2x_3)\)

It is known that

\[ V=\int_{x_1^0}^{x_1} S(x_1)\,dx_1, \]

where \(S(x_1)\) is the section of \(V(x_1x_2x_3)\) normal to the axis \(x_1\). We determine \(S(x_1)\) from (1). If the process of computing the area of a section proceeds considerably faster than integration with respect to \(x_1=t\), and there is a control circuit that makes it possible to reset the voltage at the output of the integrator computing \(S(x_1)\) to zero when \(t=2\bar{\Lambda}n\), where \(n=0,1,2\ldots m\), then we obtain

\[ V\simeq \int_{t_0}^{t}\left[\int_{0}^{2\bar{\Lambda}} r^2(t)\,dt\right]dt. \]

Fig. 2. Equipotentials of the function \(x_3=\cos x_1\cdot\cos x_2\)

The inaccuracy in computing the volume will obviously be due mainly to the number of cycles \(m\) of computing \(S\) over the interval of variation of \(x_1(t)\).

Computation of a Curvilinear Integral

A curvilinear integral of the first kind is expressed as follows:

\[ \int_h f(x_1x_2x_3)\,ds = \int_{t_0}^{t} f[x_1(t),x_2(t),x_3(t)] \sqrt{\dot r^{\,2}+r^2\dot\varphi^{\,2}+\dot x_1^{\,2}}\,dt. \]

To obtain \(\dot r\) and \(\dot x_1\), additional integrators are used, \(=\mathrm{const}\).

If one is interested only in obtaining graphical information, then it is immaterial how the functions \(x_i\) depend on time; therefore, to ensure smooth motion of the recording element (pen

DPR, the indicator beam), it is necessary to introduce stabilization of the magnitude of the linear velocity

\[ |\bar V|=\sqrt{\sum_i^n x_i^2}, \]

which can be achieved by a nonuniform variation of the angles \(\varphi_i\) in time. It should be noted that the accuracy of obtaining \(x_i(t)\) is determined mainly by the gain of the tracking amplifier and by the error in specifying \(F(x_1 x_2 \cdots x_n)\). The accuracy of all the remaining elements is not of substantial importance.

REFERENCES

1. Stefan Banach. Differential and Integral Calculus. Main Editorial Office of Physico-Mathematical Literature, “Nauka,” Moscow, 1966.

2. Leon Levine. Methods for Solving Engineering Problems Using Analog Computing Machines. “Mir,” Moscow, 1966.