Fundamentals of Constructing an Axonometric Transformation of Radiation Field Inhomogeneities

A. N. Laferov

Presented at the joint seminar of the sectors
[[unclear: DSI]] and MRD NII EFI

Investigations of a radiation field, first of all, presuppose determination of the intensity distribution in some region, that is, they include both the measurement of the field parameters at a point and the construction of a qualitative picture of the entire field.

The most widespread method of obtaining a generalized picture of the distribution is recording on X-ray film. The essential disadvantages of such recording are, first, the time gap between the course of the process and its observation, that is, the impossibility of observing the dynamics of processes; second, the visual assessment of blackening densities is limited by the brightness adaptation of the eye; third, the impossibility of correcting the sensitivity (selection of films of different contrast) during the work; and fourth, the introduction of distortions into the image during film processing.

Single-channel methods of recording give high measurement accuracy, but do not make it possible to obtain a generalized picture of the radiation field under investigation.

In this connection, combining these methods, while preserving the advantages of each, can provide more complete information about the radiation fields under investigation.

Thus, the task consists in recording the coordinates \(x, z\) of some point of the field \(D\) and the parameter \(y\) at this point, that is, in determining some function \(\psi = f(x, y, z, t)\), where \(t\) is time.

It is known that a very convenient and visual method of recording three-dimensional parameters is the axonometric representation of these parameters.

The construction of an axonometric image is based on orthogonal projection of a certain figure in a Cartesian coordinate system onto a specified projection plane; in this way, a three-dimensional object \((x, y, z)\) is reduced to a two-dimensional one \(X, Y\), while the transformation metric is preserved.

Let us consider the fundamentals of the analytic construction of an axonometric image.

Suppose that a certain body is in the Cartesian coordinate system \(x, y, z\), that is, each point of this body is defined as \(D(x_j, y_j, z_j)\). For simplicity, we shall assume that \(x, y, z\) are the coordinates of the points of this body. Projecting the system \(x, y, z\) onto the projection plane \(H\), we obtain (Fig. 1)

\[ \begin{aligned} y' &= y \cos \alpha,\\ x' &= x \cos \beta,\\ z' &= z \cos \gamma \end{aligned} \tag{1} \]

where \(\cos \alpha\), \(\cos \beta\), \(\cos \gamma\) are the distortion coefficients, that is, the angles of the coordinate axes \(x, y, z\) with the plane \(H\).

Let us define on the plane \(H\) an orthonormal coordinate system \(XY\) so that the projection axis \(y'\) coincides with the axis \(Y\); then the projections \(x'\), \(y'\), \(z'\) will be connected with the axes \(XY\) by the angles \(\omega_1, \omega_2\). After projecting the system \(x', y', z'\) onto \(XY\), we have

\[ \begin{aligned} Y &= y' - x' \cos \omega_1 - z' \sin \omega_2,\\ X &= x' \cos \omega_1 - z' \cos \omega_2 . \end{aligned} \tag{2} \]

Substituting (1) into (2), we find:

\[ \begin{aligned} Y &= y \cos \alpha - x \cos \beta \cdot \sin \omega_1 - z \cos \gamma \cdot \sin \omega_2,\\ X &= x \cdot \cos \beta \cdot \cos \omega_1 - z \cos \gamma \cos \omega_2 . \end{aligned} \tag{3} \]

The system of equations (3) is a generalized expression of the orthogonal projection of a Cartesian coordinate system onto the projection plane \(H\), in which the axis \(Y\) coincides with the projection \(y'\) of the system. For example, for isometry the angles are \(\alpha = \beta = \gamma = 35^\circ 10'\), and \(\omega_1 = \omega_2 = 30^\circ\). Substituting into (3), we obtain

\[ \begin{aligned} Y &= 0.8y - 0.4x - 0.4z,\\ X &= 0.7y - 0.7z . \end{aligned} \tag{4} \]

Figure labels: \(X\), \(Y\), \(Z\), \(x\), \(y\), \(z\), \(x'\), \(y'\), \(z'\), \(\alpha\), \(\beta\), \(\omega\), \(\omega_1\), \(\psi\).

Fig. 1

It should be borne in mind that in axonometry, as used in drafting, the coefficients for \(x\), \(y\), \(z\) are determined for ease of construction.

Thus, by specifying the angles \(\omega_1\), \(\omega_2\), that is, the type of axonometric projection, and by calculating \(\alpha\), \(\beta\), \(\gamma\), we find the values of the coefficients for \(x\), \(y\), \(z\) for constructing an axonometric image of a body specified in the coordinate system \(x\), \(y\), \(z\).

The dependences of the angles \(\alpha\), \(\beta\), \(\gamma\) on \(\omega_1\), \(\omega_2\) can be found in [1].

To realize the dependence \(\psi(x,y,z,t)\), that is, to construct an axonometry, it is necessary to specify the type of scanning of the detector in the radiation field; then the parameters \(x\), \(y\), \(z\) will be expressed as \(x(t)\), \(y(t)\), \(z(t)\), and

\[ \psi(x,y,z,t)= \begin{cases} Y = 0.8y(t)-0.4x(t)-0.4z(t),\\ X = 0.7y(t)-0.7z(t), \end{cases} \tag{5} \]

where \(x(t)\) and \(y(t)\) determine the location of the detector in the radiation field \(D(x(t),z(t))\), with \(y(t)\) being the field parameter at point \(D\).

By applying to the \(Y\) axis of an oscilloscope, or of any other two-coordinate recorder, the algebraic sum of the three terms according to (5), and to \(X\) the sum of the two terms, on the screen we obtain an isometric projection of a region of the field of length \(x_{\max}\), width \(z_{\max}\), and with the parameter being the intensity.

The hardware implementation of this recording method is described in [2].

Literature

1. Laferov A. N., Naats I. E., Bylino N. M. Construction of axonometric images on the Minsk-2 digital computer. Izvestiya TPI, 1970.

2. Vorob’ev V. A., Laferov A. N., Filonin O. V. A device for constructing an axonometric image of a signal on the screen of a cathode-ray tube when measuring inhomogeneities of the spatial distribution of radiation fields. (The present collection.)