Perspective Representation of Computation Results Using an Analog Computer

This is an English translation of the original German-language document “Perspektivische Darstellung von Rechenergebnissen mit Hilfe eines Analogrechners” published by AEG-Telefunken (Datenverarbeitung division), document ASD 051 0469.

Perspective Representation of Computation Results Using an Analog Computer

In many areas of technology and natural science, the need arises to represent results in the form of a surface z = f(x, y) in three dimensions. This representation can make dependencies between two variables graphically clear. For example, if the solution curves of a system of differential equations in two variables are computed, they can be plotted on a two-axis oscilloscope (X-Y recorder). However, to display the dependence on a third variable, z, in addition, one must find a way to produce a perspective (3D) representation.

In general, one calculates for the perspective representation at a viewing angle φ, the screen coordinates:

x’ = x + z · cos φ
y’ = y + z · sin φ

This formula implies that for a spatial 3D representation, one needs a circuit that adds a fraction of z (specifically z · cos φ or z · sin φ) to the original x and y values.

Fig. 1: Block diagram of the perspective representation of a curve using an analog computer

The simple “3D circuit” described here achieves this through the use of two multipliers. The product cos φ and sin φ can be set by potentiometers to a constant angle φ.

Circuit Variants

Fig. 2: 3D circuit with angle φ settable by potentiometer

The two sine/cosine values are generated as static constants via potentiometers. In this configuration, the viewing angle φ is fixed manually before the run.

Fig. 3: 3D circuit with time-varying angle φ

In the following circuits (Figs. 3–5), the rotation angle φ can change continuously on its own, so that on the oscilloscope one gains the impression of a rotating three-dimensional object.

Fig. 4: 3D circuit with time-varying angle φ (rotation can be stopped)

This variant adds a HALT/DREHEN (Stop/Rotate) control to freeze the current viewing angle at any point during the rotation.

Fig. 5: 3D circuit with time-varying angle φ (rotation can be stopped and reversed)

An extended version adds a VOR/ZURÜCK (Forward/Backward) control, so the rotation direction can also be reversed, in addition to being stopped.

In a similar manner, somewhat more complex equations and circuits can be set up that allow projection onto two or three axes, to gain a spatial impression of the curve trace.

Simple Example of a Spatial Curve with Three Variables u, v, t

Given the following system of differential equations:

du/dt = −d·u − c·v (1)
dv/dt = u (2)

with initial conditions u(0) = a; v(0) = 0.

Fig. 6: Analog circuit for the above differential equations

Solutions of the differential equations:

v = (a/ω) · sin(ωt) · e^(−d/2 · t)
u = a · √(1 + d²/(4ω²)) · cos(ωt − α) · e^(−d/2 · t)
with ω = √(c − d²/4), α = arctan(d / 2ω)

2D Special Cases of the 3D Perspective Display

In the perspective representation, the special cases for φ = 0° and φ = 90° yield the two-dimensional special-case projections (see Fig. 7):

At φ = 0°: the x–y plane shows u and v, yielding a damped spiral in the phase plane.
At φ = 90°: the x–t plane shows a damped oscillation of u (or v) as a function of time t.

Fig. 7: Two-dimensional representations as special cases of the perspective display of a spatial spiral

Perspective Display of a Spatial Spiral at Various Viewing Angles

Fig. 8: Perspective representation of a spatial spiral at various rotation angles φᵢ, with φᵢ₊₁ > φᵢ

Eight views are shown, stepping through a range of rotation angles, illustrating how the same 3D spiral appears from different perspectives — ranging from a nearly “edge-on” view (resembling a damped 1D oscillation) to nearly “face-on” (resembling a compact 2D spiral), and various intermediate oblique views.

[Translation covers all 7 pages of the original document.]