CONSTRUCTION OF AXONOMETRIC PROJECTIONS ON A DIGITAL COMPUTER

N. M. Bylino, A. N. Laferov, I. E. Naats

(Presented by the scientific seminar of the Department of Higher Mathematics, TPI)

The use of computing machines for automating the construction of graphic images leads to the need to develop methods for encoding geometric information. This article proposes one method for encoding and processing geometric information in the construction of axonometric images. The method considered in [1] for encoding a graphic image by means of a set of adjacent points, while general, has considerable redundancy. In a number of practical cases, however, a more economical method for encoding a graphic image may be proposed, namely: specifying lists of coordinates of certain points and of the connections between them. In particular, for a polygon it is convenient to take, as such lists, the lists of vertices and edges. The intermediate points required when detailing the construction of axonometric images can be obtained by linear interpolation, which significantly reduces the volume of input information.

The apparatus of axonometric projection is determined by three angles \(\omega_1, \omega_2, \omega_3\) (i.e., the angles between the projections of the axes of the system in which the geometric body is specified; Fig. 1). The values of the angles \(\omega_1, \omega_2, \omega_3\) are determined [2], under orthogonal projection, by the angles \(\alpha, \beta,\) and \(\gamma\), which in turn characterize the position of the original coordinate system relative to the plane of projections,

\[ \sin \omega_1=\frac{\cos\alpha}{\sin\beta\cdot\sin\gamma}, \]

\[ \sin \omega_2=\frac{\cos\beta}{\sin\alpha\cdot\sin\gamma}, \]

\[ \sin \omega_3=\frac{\cos\gamma}{\sin\alpha\cdot\sin\beta}. \tag{1} \]

Since the angles \(\omega_1, \omega_2, \omega_3\) are known, then, carrying out the inverse transformations, we obtain

\[ \cos \alpha = \sqrt{ \frac{\cos^2\omega_1+\cos\omega_1\cos\omega_2\cos\omega_3} {\sin^2\omega_2-\sin^2\omega_1+\cos^2\omega_2\sin^2\omega_3} }, \]

\[ \begin{gathered} \cos\beta=\sqrt{\frac{\cos^2\omega_2+\cos\omega_1\cos\omega_2\cos\omega_3} {\sin^2\omega_3-\sin^2\omega_2+\cos^2\omega_3\sin^2\omega_1}},\\ \cos\gamma=\sqrt{\frac{\cos^2\omega_3+\cos\omega_1\cos\omega_2\cos\omega_3} {\sin^2\omega_1-\sin^2\omega_3+\cos^2\omega_1\sin^2\omega_2}}. \end{gathered} \tag{2} \]

These expressions are valid for \(\omega_1,\ \omega_2,\ \omega_3 \ge 90^\circ\). For any \(\omega=90^\circ\), the axonometric image reduces to Monge projections.

If \(X'Y'Z'\) are the coordinates of a point of the Cartesian system, then the system of equations for axonometric construction will take the form

\[ \begin{gathered} x=X'\cos\alpha,\\ y=Y'\cos\beta,\\ z=Z'\cos\gamma. \end{gathered} \tag{3} \]

The transition from (3) to a plane with two Cartesian axes, necessary for simplifying the process of constructing the axonometric image, is carried out according to the expressions

\[ \begin{gathered} X=y\sin\omega_1+x\sin(\omega_1+\omega_2),\\ Y=z+y\cos\omega_1+x\cos(\omega_1+\omega_2), \end{gathered} \tag{4} \]

where \(X, Y\) are the abscissa and applicate in the drawing plane, and the \(Y\)-axis of the drawing coincides with the axonometric axis \(Z\).

In practice, a geometric solid cannot always be specified by an ordered list of coordinates. More often there are three Monge projections. It is then necessary to enter the initial data on the solid from the Monge projections into the machine and, from them, compute the coordinates of the axonometric image of the given solid.

Fig. 1

Let us consider in more detail the construction of the axonometric projection of a solid onto some plane, if the solid is given in three Monge projections. As such a solid we shall take a cube with edge \(a=1\) (Fig. 1). From the Monge projections, the set of vertices of the cube can be specified by two lists of coordinates of its vertices, compiled in arbitrary order and corresponding to any pair of planes \((H, V, W)\). From these two lists, corresponding to the projections of the vertices, a common list corresponding to the actual set of vertices of the cube is formed by selecting pairs with one common coordinate (say \(X\)). For the present example the resulting list will have the form \(a, b, c, d, e, f, k, l\).

The determination of the actual connections between the vertices of a solid is carried out as follows. Let us take, for example, vertex \(K\). From one Monge projection one can compile a list of possible connections of this point with the other vertices of the cube. Compiling such lists for all three projection planes, we obtain the table

\[ \begin{aligned} K_v&\ d\ b\ e\ f\ l\\ K_h&\ d\ b\ e\ a\ l\\ K_w&\ d\ a\ e\ f\ l, \end{aligned} \]

where \(K_v\), \(K_h\), and \(K_w\) conventionally denote the projections of point \(K\) onto the corresponding planes. It is clear from the table that point \(K\) has connections with points \(d\), \(e\), and \(l\), since an actual connection (edge) must be present in all three Monge projections. Having performed the indicated operations for each vertex, we obtain a list of connections that determines the configuration of the solid.

After the actual connections between vertices have been established, the ordering of the array of vertex coordinates is carried out according to the following rule:

1. Beginning with the first pair in the list of connections, we successively compare it with the remaining ones and find a pair that has a vertex in common with the initial pair \((Kl, lf)\).

2. The second vertex of the pair found is similarly compared with the remaining ones until a pair is encountered in the list with a vertex that has the desired coordinates \((kl, ef, ek, dk, db, cb, ac, ae)\).

3. Next the following pairs (edges) are taken, and the same procedure is carried out \((ad, cb, bl)\).

Fig. 2

Simultaneously with the search, an ordered recording of the sequence of vertices found is performed, and thus these vertices are excluded from the original list of connections.

After the indicated operations, the axonometric projection of the solid is determined according to the formulas given above.

The results of constructing two axonometric projections on the “Minsk-1” computer are shown in Fig. 2. Intermediate points of the edges were obtained by linear interpolation with a step of 0.2 of the smallest edge of the solid being depicted.

The indicated method can be applied to geometric solids of more complex configuration.

Conclusions

1. In constructing axonometric images using a digital computer, the paper proposes introducing the initial graphic information from Monge projections in the form of unordered lists of projections of vertices (in the general case, a set of defining points) and of possible—that is, observed in the projections—connections between them.

1. The algorithm described in this work orders these lists and identifies the actual connections between vertices (edges).

2. The intermediate points on the edges, needed for detailing when constructing an axonometric image, are proposed to be obtained by linear interpolation over the ordered list of vertices, which substantially reduces the amount of initial geometric information entered into the machine.

References

1. S. A. Frolov. Cybernetics and Engineering Graphics. Moscow, 1967.
2. Geometry. EEM, vol. 4. Moscow, 1963.