MODELING THE EQUATIONS OF PROJECTION OSCILLOGRAPHY ON THE “EMU-10” MACHINE

I. E. NAATS, V. M. REIDER

(Presented by the scientific seminar of the Computing Laboratory of TPI)

It is known that, in ordinary oscillography of electrical processes, the oscillogram is a curve in the plane of the tube screen, determined by a pair of points \((x, y)\). However, using projection methods, one can construct a projection image of a spatial curve on the screen plane. The fundamentals of the method of projection oscillography are set forth in works [1, 2].

Fig. 1. System of spatial geometric coordinates

In the present article it is proposed to use an electronic modeling machine for constructing projection oscillograms that represent various functional dependences in the space of three coordinates \((x, y, z)\). To obtain a “spatial” oscillogram on the oscilloscope screen, it is necessary to find the relationship between the coordinates of a point in space \((x, y, z)\) and the coordinates of the projection of this same point onto the plane \((X, Y)\).

Let us consider the simplest case, when one of the axes of the spatial coordinate system, the \(y\)-axis, is aligned with the \(Y\)-axis of the plane coordinate system (Fig. 1). It is not difficult to find the required relations:

\[ \left\{ \begin{aligned} X &= x \cos \varphi - z \cos \psi,\\ Y &= y - z \sin \psi - x \sin \varphi . \end{aligned} \right. \tag{1} \]

In these equalities, \(\varphi\) is the angle of “rotation” of the point about the \(y\)-axis, and \(\psi\) is the angle of “rotation” of the point about the \(x\)-axis.

Thus, knowing the trajectory of motion of some point in space \((x, y, z)\), one can obtain the projection of the trajectory of motion of this point onto the plane \((XY)\).

The transition from the spatial coordinate system to the plane one, according to (1), can be carried out with the aid of the analog computer “EMU-10.” The functional diagram is presented in Fig. 2, in which the following are denoted:

1 — functional unit,
2 — generator of the sweep voltage along the \(x\)-axis,
3 — generator of the sweep voltage along the \(z\)-axis,
4 — unit of sine-cosine potentiometers for rotation of the body about the \(y\)-axis,
5 — unit of sine-cosine potentiometers for rotation of the body about the \(x\)-axis,
6 — adder of the components of the \(Y\)-axis,
7 — adder of the components of the \(X\)-axis,
8 — oscilloscope screen.

Fig. 2. Functional diagram of the model

To obtain the projection image of a three-dimensional body on the oscilloscope screen, it is necessary that the frequency of the generator of the sweep voltage along the \(x\)-axis be at least an order of magnitude higher than the frequency of the generator of the sweep voltage along the \(z\)-axis.

The voltages of these generators enter the functional unit, from whose output a function of two variables \(y = f(x, z)\) is fed to the adder of the components of the \(Y\)-axis. The generators of the arguments “\(x\)” and “\(z\)” are built on ordinary integrators. The unit of sine-cosine potentiometers consists of two potentiometers connected by one shaft; the output values of one of them vary according to a sine law, and of the other according to a cosine law. When the shaft of the potentiometers of the angle \(\varphi\) is rotated, the body “rotates” about the \(y\)-axis; when the shaft of the potentiometers of the angle \(\psi\) is rotated, the body “rotates” about the \(x\)-axis. The “rotation” of the body means a change in the viewing point of the given spatial curve or surface.

Fig. 3. Projection oscillograms with successive rotations about axes.

As the body of revolution, a surface described by the equations was taken:

\[ \left\{ \begin{aligned} y&=-1 &&\text{for } -\infty < x < -1,\\ y&=Ax, && -1 \leq x \leq 1,\\ y&=1, && 1 < x < \infty,\\ z&=k\Delta z, \end{aligned} \right. \tag{2} \]

where \(A\) is a constant, \(\Delta z\) is the sweep step (\(k\) is the number of sweep cycles). Figure 3 presents three oscillographic projections of the surface (2) for different positions in space.

Conclusions

Using the above-described method of spatial oscillography, with the aid of a modeling machine it is possible to obtain on the screen of an oscilloscope any surface or three-dimensional body; moreover, rotation of this body (surface) is possible, i.e., a change in the viewing point of the given object.

LITERATURE

1. L. A. Zubritskii. Scientific Notes. Publishing House of Kharkov University, vol. 105, 1957.
2. “Journal of Applied Physics.” Vol. 18, No. 9, September, 1947, pages 819–829.
3. Abramson. The Modern Cathode Oscillograph, part II, 1951.