QUESTIONS OF CONSTRUCTING PROJECTION IMAGES IN THE REGISTRATION OF GRAPHICAL INFORMATION

I. E. NAATS

(Presented by the scientific seminar of the computing laboratory)

At the present time, questions of processing and visually presenting graphical information processed by means of computing machines are acquiring ever greater importance. In this connection, certain difficulties are caused by the registration of three-dimensional information on the two-dimensional surface of a registration document (paper tape, the screen of a cathode-ray tube, etc.). Some methods for solving this problem may be found in works [1, 2].

One of the possible variants of graphical registration of three-dimensional information on a plane carrier is the construction, on the projection plane, of projection images of spatial lines, surfaces, and bodies. The use of projection methods proves especially convenient for the output of information from analog computing machines when solving three-dimensional problems, since in this case the solution can be obtained in the form of three continuous functions of time \((x(t), y(t), z(t))\), describing the motion of a certain point \(A\) with coordinates \((x, y, z)\) in three-dimensional space [3]. With this form of information output, direct graphical registration is possible by means of self-recording instruments and cathode-ray indicators. In turn, the motion of point \(A\) with coordinates \((x, y, z)\) may be regarded, from the geometric point of view, as motion over the surface \(z(x, y)\). To register the relief of this surface while preserving visual clarity on a plane carrier, projection methods may be used.

Works [4, 5], in particular, are devoted to the application of the indicated methods for purposes of graphical registration of three-dimensional information; there, primary attention is given to questions of constructing so-called free projection images [6]. In a free image, the visual clarity of the plane image of the given body is important, while the position of the original in space is entirely unimportant. Accordingly, the parameters characterizing the projection apparatus remain unknown.

When geometric information is output from a machine and, in particular, when physical processes are modeled on an analog computer, it becomes necessary, from the obtained projection image of the function \(z(x, y)\), to determine the position of the original in space. In this case it is necessary to construct so-called rigid images [6], for which all parameters characterizing the projection apparatus are known.

With a rigid image it is possible to determine the position of the original in space and the metric relations in it.

In the present work we consider questions of constructing rigid projection images for the purpose of extracting geometric information from a computer, represented in the form of functions of two variables \(z(x,y)\).

A projection image of some body (surface, line), whose spatial position is determined in some coordinate system \(S\), on a plane \(W\) is obtained as the result of projecting this body onto the indicated plane.

The most widespread methods of constructing images are parallel and central projection. The projection method determines the form of the relations connecting the coordinates of some point \(A(x,y,z)\) in the system \(S\) with the coordinates \((X,Y)\) of the projection of this point on the plane \(W\). To derive the projection relations it is necessary, in addition to the projection method, to specify the position of the system \(S\) relative to the projection plane \(W\). In general, all this determines the projection apparatus. The position of the system \(S\) may be considered fixed if the position of the origin of coordinates and the direction of the axes of the system relative to the plane \(W\) are specified.

Fig. 1. Geometric constructions for deriving projection relations

For simplicity we shall assume that the origin of coordinates of the system \(S\), the point \(O\), lies in the plane \(W\); the direction of the axes \(x,y,z\) will be specified, respectively, by the angles \(\alpha,\beta,\gamma\) (Fig. 1). Let \(\vec n\) denote the normal to the plane \(W\) at the point \(O\). The direction of the vector \(\vec n\) in the system \(S\) will be determined by the angles \(\alpha',\beta',\gamma'\), where

\[ \left. \begin{aligned} \alpha'&=\frac{\pi}{2}-\alpha,\\ \beta'&=\frac{\pi}{2}-\beta,\\ \gamma'&=\frac{\pi}{2}-\gamma. \end{aligned} \right\} \tag{1} \]

For the system of angles (1) the following equality is valid:

\[ \cos^2 \alpha' + \cos^2 \beta' + \cos^2 \gamma' = 1. \tag{2} \]

Substituting (1) into (2), we find an analogous relation for the angles \(\alpha, \beta, \gamma\) characterizing the projection apparatus,

\[ \sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma = 1. \tag{3} \]

Relation (3) shows that, of the angles \(\alpha, \beta, \gamma\), only two angles are independent parameters; therefore, in what follows we shall pass to two independent parameters. As such parameters it is convenient to choose the angle of inclination of any one of the axes of the system \(S\) to the plane \(W\), and the angle of rotation of the system \(S\) about this axis relative to some fixed position. Figure 1 shows the rotation of the system \(S\) relative to the axis \(x\); here the angle of rotation \(\nu\) is defined as the angle between the axis \(y\) and some fixed position of it, \(y_0\). The vector (axis) \(y_0\) lies in the plane \(H\), containing the axis \(x\) and the vector \(\vec n\). The angle \(\nu\) also determines the rotation of the axis \(z\) from its initial position \(z_0\), lying in the projection plane \(W\). On the basis of the indicated constructions one may choose a coordinate system on the plane \((X,Y)\). We choose the direction of the axis \(X\) along the line of intersection of the planes \(W\) and \(H\); the axis \(Y\) is determined accordingly. It is now necessary to express the angles \(\beta\) and \(\gamma\) in terms of the parameters \(\alpha\) and \(\nu\). The corresponding geometric constructions are indicated in Fig. 1, whence it is not difficult to find that

\[ \sin \beta = \cos \nu \cdot \cos \alpha . \tag{4} \]

Similarly,

\[ \sin \gamma = \sin \nu \cdot \cos \alpha . \tag{5} \]

By direct verification we see that (4) and (5) satisfy (3); thus the parameters \(\alpha\) and \(\nu\) completely determine the rotation of the system \(S\) in space relative to the point \(O\), while the axis \(X\) does not leave the plane \(H\). We note that the indicated two motions (in \(\alpha\) and \(\nu\)) are quite sufficient for a convex body specified in the system \(S\) to be completely viewed from a given direction, for example in the direction of the normal. Or, in another formulation: the indicated rotations are sufficient for the entire surface of a convex body to be illuminated from any direction [7].

In parallel projection, the selected parameters \(\alpha\) and \(\nu\) are not entirely sufficient for characterizing the projection apparatus. The metric parameter \(l\), which determines the distance of the point \(O\) to the plane along the normal, in this case does not affect the form of the projection relations.

Let us determine the position of the projections of the axes of the system \(S\) on the plane \(W\), \(x'\), \(y'\), \(z'\), relative to the system \((X,Y)\), by the angles \(\varepsilon\) and \(\xi\) (Fig. 1). Two angles are quite sufficient, since \(x'\) is always projected onto the direction of the axis \(X\).

Using the constructions in Fig. 1, it is not difficult to show that

\[ \sin \varepsilon = \frac{\sin \nu}{\sqrt{1-\cos^2 \nu \cdot \cos^2 \alpha}} . \tag{6} \]

Similarly,

\[ \sin \xi = \frac{\cos \nu}{\sqrt{1-\sin^2 \nu \cdot \cos^2 \alpha}} . \tag{7} \]

Using (6) and (7), for parallel projection one can write

\[ \left\{ \begin{aligned} X&=-x\cos\alpha+y\cos\nu\cdot\sin\alpha+z\sin\nu\cdot\sin\alpha,\\ Y&=y\sin\nu-z\cos\nu. \end{aligned} \right. \tag{8} \]

Relations (8) determine the coordinates of the projection of a point \((x,y,z)\) from the system \(S\) onto the plane \(W\) for given projection parameters \(\alpha\) and \(\nu\). Conversely, knowing the values \((X,Y)\) of the point \((x,y,z)\), relations (8) determine the values of \(\alpha\) and \(\nu\), and thereby the position of the system \(S\) (respectively, of the original) in space relative to the projection plane \(W\). As a result, a rigid image of the original on the plane \(W\) is realized.

In some cases it is desirable to obtain an image of the original with the projections of the axes of the system \(S\) plotted on the plane \(W\); here the angles between the projections of the axes must assume a definite value, i.e., the problem of constructing an isometry arises. One of the common examples of such a problem is the construction of an orthogonal isometry, for which

\[ (\widehat{x',y'})=(\widehat{x',z'})=(\widehat{y',z'})=120^\circ . \]

For this case the angles in the projection plane \(\varepsilon\) and \(\zeta\) are equal to \(60^\circ\). Using (6) and (7), one can determine the orientation of the system \(S\) in space. Indeed, with respect to the parameters \(\alpha\) and \(\nu\) we have the system

\[ \frac{\sin\nu}{\sqrt{1-\cos^2\nu\cdot\cos^2\alpha}} = \frac{\cos\alpha}{\sqrt{1-\sin^2\nu\cdot\cos^2\alpha}} = \frac{\sqrt{3}}{2}. \tag{9} \]

Solving (9), we find

\[ \cos\alpha=\sqrt{\frac{2}{3}}, \]

\[ \cos\nu=\frac{1}{\sqrt{2}} \]

and, respectively, \(\alpha=35^\circ\) and \(\nu=45^\circ\). Other axonometric systems of axes \(x',y',z'\) can be constructed analogously. The application of the results obtained to the construction of projection graphical images of functions \(z(x,y)\) is set forth in [8].

REFERENCES

1. A. F. Temnikov. Automatic recording instruments. Mashgiz. Moscow, 1960.

2. V. I. Rybak, A. N. Shishonok. Indicator device for outputting observation results from a machine. “Avtomatika i priborostroenie,” No. 1, 1963.

3. V. A. Tartakovskii. Construction on a computer of lines lying on \(n\)-dimensional surfaces. (The present collection).

4. M. A. Parker, E. P. R. Wallis. The proceeding Instit. of Elect. Eng. vol. 95, part III, No. 37, 1948.

1. P. R. Wallis. Journal of Appl. Phys. vol. 18, No. 9, Sept. 1947, pp. 818–829.
2. Geometry IV. EEM. Fizmatgiz, Moscow, 1963.
3. V. G. Boltyanskii. Theorems and Problems of Combinatorial Geometry. “Nauka,” 1966.
4. I. E. Naatz, V. M. Reider. Construction on an AVM of projection images of spatial figures by the method of central projection. (This collection).