CONSTRUCTION ON AN ANALOG COMPUTER OF PROJECTION IMAGES OF SPATIAL FIGURES BY THE METHOD OF CENTRAL PROJECTION

I. E. NAATS, V. M. REIDER

(Presented by the scientific seminar of the computing laboratory)

This paper considers questions of obtaining projection images of three-dimensional geometric objects on an analog computer, with registration on the screen of a cathode-ray indicator. The main attention is devoted to the method of central projection in constructing images on the surface of a flat carrier and to comparing the obtained oscillograms with oscillograms corresponding to parallel projection.

One of the conveniences, in addition to those noted in [1], of using projection methods in registering geometric information on an analog computer is the possibility of setting the projection relations as part of the program of the problem being solved. As a result, the output quantities \(X\) and \(Y\) can be fed to a two-coordinate recorder or to a cathode-ray indicator for registering the projection image of the solution in the form \(z(x,y)\).

It is known that projection images of geometric objects obtained by the method of central projection possess greater visual clarity in comparison with analogous images obtained by the method of parallel projection. However, in this case the image loses in metricity, i.e., the determination of quantitative relations from the image becomes more complicated. Of definite interest is a comparison of these two methods using images of a surface \(z(x,y)\) as an example. We shall first make several remarks concerning the derivation of projection relations in central projection, using the projection system of [1].

It is quite obvious that, in passing to central projection, the angular parameters \(\alpha\) and \(\nu\) of the projection apparatus must be supplemented by metric characteristics; as such we shall choose the distances of the projection pole \(P\) and of the origin of coordinates of the system \(S\), the point \(O\), to the projection plane \(W\) (Fig. 1). For simplification we shall assume that the pole \(P\) and the point \(O\) lie on the normal \(\vec n\) to \(W\). In the particular case, as shown in Fig. 1, the point \(O\) falls on the plane \(W\); then the corresponding parameter \(l=0\). In contrast to parallel projection, the parameter \(l \ne 0\) in central projection and will enter into the projection relations.

Let there be a point \(A\) in the system \(S\) with coordinates \(x, y, z\) (the same vector \(\vec r\) with direction angles \(\varphi, \psi, \vartheta\)). To find

of the projection relations it is necessary to determine the projection of the vector \(\vec r\) onto the axes \(X, Y\) of the plane \(W\). To this end we first project the vector \(\vec r\) onto the plane \(W\). The magnitude of this projection is determined by the expression

\[ OA'=\frac{L\cdot r\cos\omega}{L-l-r\sin\omega}, \tag{1} \]

where

\[ \begin{aligned} r&\text{ is the modulus of }\vec r,\\ L&\text{ is the distance from the pole }P\text{ to the plane and} \end{aligned} \]

Fig. 1. Geometric constructions in the central projection of a point onto a plane

\[ \sin\omega=\sin\alpha\cdot\cos\varphi+\sin\beta\cdot\cos\psi+\sin\gamma\cdot\cos\vartheta. \tag{2} \]

Then the projection of the segment \(OA'\) onto the axes \(X\) and \(Y\) is determined:

\[ \left. \begin{aligned} X&=OA'\cdot\cos\Omega\\ Y&=OA'\cdot\sin\Omega \end{aligned} \right\}. \tag{3} \]

Having carried out the corresponding geometric constructions, it is not difficult to find the value of the angle \(\Omega\):

\[ \left. \begin{aligned} \sin\Omega&=\frac{\cos\psi\cdot\sin\gamma-\cos\vartheta\cdot\cos\gamma}{\cos\omega},\\[4pt] \cos\Omega&=\frac{\cos\psi\cdot\cos\gamma\cdot\sin\alpha+\cos\vartheta\cdot\sin\alpha\cdot\sin\gamma-\cos\alpha\cdot\cos\varphi}{\cos\omega}. \end{aligned} \right\} \tag{4} \]

Substituting (4) into (3), after elementary transformations we obtain the projection relations connecting the coordinates of the point \((x,y,z)\) with

Fig. 2. Projection oscillograms obtained by the method of parallel projection.

Fig. 3. Projection oscillograms obtained by the method of central projection.

coordinates of its projected central image on the plane:

\[ \left. \begin{aligned} X&=L\cdot \frac{y\cos\nu\cdot\sin\alpha+z\sin\alpha\cdot\sin\nu-x\cos\alpha} {L-l-x\sin\alpha-y\cos\alpha\cdot\cos\nu-z\cos\alpha\cdot\sin\nu},\\ Y&=L\cdot \frac{y\sin\nu-z\cos\nu} {L-l-x\sin\alpha-y\cos\alpha\cdot\cos\nu-z\cos\alpha\cdot\sin\nu}. \end{aligned} \right\} \tag{5} \]

It is interesting to note that, when the pole \(P\) is removed infinitely far from the plane \(W\), the projected image of a point, according to (5), differs less and less from the projected image of this point under parallel projection. In other words, as \(L\to\infty\), relations (5) pass into the relations of parallel projection [1], namely:

\[ \left. \begin{aligned} X&=y\cos\nu\cdot\sin\alpha+z\sin\alpha\cdot\sin\nu-x\cos\alpha,\\ Y&=y\sin\nu-z\cos\nu. \end{aligned} \right\} \tag{6} \]

The obtained system of relations (5) is sufficiently simple both for computation and for modeling, in contrast to the equations proposed in [2], where the parameters of the projection apparatus were determined by specifying the projection plane \(W\) and the pole \(P\) in the system \(S\), despite a number of significant simplifications introduced by the author into the projection apparatus.

Using relations (5) and (6), one can carry out a comparative analysis of the projection images obtained in this way, using as an example some second-order surface.

In Figs. 2 and 3 projection oscillograms are presented, giving images of a cylindrical surface (also of a helical line), respectively in parallel and central projection on the screen of a cathode-ray tube. The set of problems for synthesizing the surface and the projection relations was carried out on the MN-14 analog computer with the following data: \(\nu=0\) and \(\alpha=60^\circ, 45^\circ, 30^\circ\) (from top to bottom in Figs. 2, 3). The oscillograms shown make it possible to judge the visual quality of the projection images. The oscillograms of Fig. 3 are more pictorial, although from them it is difficult to judge the linear dimensions of the cylinder. However, this is not a substantial drawback of these oscillograms, since by purely instrumental means one can introduce a system of coordinate marks for determining metric relations in the original. The described methods for constructing projection oscillograms can also be used for recording more complex spatial figures.

REFERENCES

1. I. E. Naats. Questions of constructing projection images in the recording of graphical information. (This collection).

2. A. V. Kuznetsov. Nomography of constructions of central axonometric projections. See: Nomographic Collection No. 2. Computing Center of the USSR Academy of Sciences. Moscow, 1964.