SIMPLIFIED AXONOGRAPH

B. A. Mashukov, L. S. Skripov

(Presented by the scientific seminar of the Department of Descriptive Geometry and Graphics).

In the period from 1940 to 1961, quite a large number of designs were proposed for devices for the mechanical construction of axonometric images, so-called “axonographs.” All of them are theoretically constructed correctly, but they have a rather complex design, which, apparently, has been an obstacle to their widespread use in drafting practice. Some axonograph designs construct axonometric images with exact allowance for the distortion coefficients along the axonometric axes and, consequently, produce images from which it is difficult to determine the dimensions of the object in nature. Naturally, the idea arises of creating an axonograph which, while simple in design, could construct axonometric images at the scales recommended by the All-Union Standards for drawings in machine building.

In the present article a design is proposed for a simplified axonograph, the theory of which was developed by Docent L. S. Skripov, while its external design and study were carried out by B. A. Mashukov, assistant of the Department of Descriptive Geometry and Graphics.

The design of the device obtained proved analogous to the design proposed by N. D. Biryuchevsky, a staff member of the Siberian Metallurgical Institute, in his article “The Projective Theory of Devices for Drawing Perspective and Axonometric Images.” However, the authors of the present article constructed their device on the basis of other theoretical considerations, independently of N. D. Biryuchevsky, and carried out a more profound study in its application to practical problems, which is not found in N. D. Biryuchevsky’s article.

The scheme of the proposed device and the theory of its operation are presented in Fig. 1 in the projection of the simplest geometric solid—a cube. In the left part of the drawing are shown the frontal and horizontal projections of the cube 12345678. The horizontal lines are the edges of the drawing board, along which horizontal rulers I and II slide. To ruler-slide II, inclined rulers IV and VI are attached on hinges; they can move only simultaneously and can be fixed on the slide at different distances from one another. To the lower ruler-slide I, inclined rulers III and V are attached on hinges; these likewise can move only simultaneously. The angles of inclination of the rulers \(\varphi_0\), \(\psi_0\), \(\varphi_3\), \(\psi_3\) may have different values depending on the axonometric image being constructed.

The left rulers III and IV are set on the horizontal and frontal projections of some point, while the right rulers V and VI must intersect at the point showing its axonometric projection.

Thus all points of the projection of the given object are traversed, and their axonometric images are obtained, which are then joined by the corresponding straight or curved lines, taking into account their visibility in the projections.

In Fig. 1 the edges of the cube are arranged parallel to the coordinate axes: edge 1—4 is parallel to the axis \(OX\), edge 1—2 is parallel to the axis \(OY\), and edge 1—5 is parallel to the axis \(OZ\), and are denoted by the letters \(A, B, C\).

Suppose that these edges are projected into the magnitudes \(A_1, B_1, C_1\); then the distortion coefficients along the axonometric axes will be expressed by the ratios

\[ \frac{A_1}{A}=p,\qquad \frac{B_1}{B}=q, \]

\[ \frac{C_1}{C}=r. \]

Fig. 1

Since the proposed instrument is designed for constructing only rectangular axonometry (isometry and dimetry) and frontal oblique dimetry, the distortion coefficients must have the following values (according to GOST):

\[ \text{for isometry } p=1,\ q=1,\ r=1, \]

\[ \text{for dimetry } p=1,\ q=\frac{1}{2},\ r=1, \]

\[ \text{for frontal dimetry } p=1,\ q=\frac{1}{2},\ r=1. \]

Since in all these cases the coefficient \(p=1\), the cube edge 1—4 \((A)\) must be projected in axonometry without distortion. Consequently, if ruler III is moved by the amount \(a=A\), then ruler V must also move by the amount \(a=A=A_1\), and then we obtain

\[ \frac{A_1}{A}=p=1. \tag{1} \]

When rulers III and IV are set at points 2 and 5, the points of their attachment on sliders I and II will move by the amounts \(b\) and \(c\). Then from the triangles 1—2—9 and 1—5—10 one can obtain

\[ b=B\cdot \operatorname{ctg}\varphi_0 \qquad \text{and} \qquad c=C\cdot \operatorname{ctg}\psi_0. \tag{2} \]

Let us establish the dependence between the angles of inclination of the slider-rulers to rulers III, IV, V, VI, i.e. the angles \(\varphi_0, \psi_0, \varphi_3, \psi_3\), under the condition that the cube edges \(B\) and \(C\) in axonometry receive the magnitudes \(B_1\) and \(C_1\).

From triangle \(1—2—11\) (right-hand drawing) we have

\[ \frac{b}{\sin\{180^\circ-(\varphi_3+\psi_3)\}} = \frac{B}{\sin\varphi_3} = \frac{b}{\sin(\varphi_3+\psi_3)}, \]

or

\[ \frac{B_1}{b} = \frac{\sin\varphi_3}{\sin(\varphi_3+\psi_3)}, \]

and since \(\dfrac{B_1}{B}=q\), then

\[ \frac{q\cdot B}{b} = \frac{\sin\varphi_3}{\sin(\varphi_3+\psi_3)}, \]

or, from expression (2), we have

\[ \frac{q\cdot B}{B\cdot \operatorname{ctg}\varphi_0} = \frac{\sin\varphi_3}{\sin(\varphi_3+\psi_3)}, \quad \text{or} \quad \frac{q}{\operatorname{ctg}\varphi_0} = \frac{\sin\varphi_3}{\sin(\varphi_3+\psi_3)}. \tag{3} \]

From triangle \(1—5—12\) we have

\[ \frac{c}{\sin\{180^\circ-(\varphi_3+\psi_3)\}} = \frac{c_1}{\sin\psi_3}, \]

or

\[ \frac{c}{C_1} = \frac{\sin(\varphi_3+\psi_3)}{\sin\psi_3} \]

or

\[ \frac{c}{C\cdot r} = \frac{\sin(\varphi_3+\psi_3)}{\sin\psi_3}. \]

Taking expression (2) into account, we have

\[ \frac{C\cdot \operatorname{ctg}\psi_0}{C\cdot r} = \frac{\sin(\varphi_3+\psi_3)}{\sin\psi_3}, \quad \text{or} \quad \frac{\operatorname{ctg}\psi_0}{r} = \frac{\sin(\varphi_3+\psi_3)}{\sin\psi_3}. \tag{4} \]

We determine the magnitudes of the angles \(\varphi_0, \psi_0, \varphi_3, \psi_3\) for obtaining isometry, dimetry, and frontal dimetry.

1. Isometry. We take the distortion coefficient to be equal to

\[ p=q=r=1. \]

On the basis of the condition adopted above, that in isometry the edge of the cube \(1—4=A\) must be arranged horizontally, the isometric image of the cube will be turned clockwise through an angle of \(30^\circ\) and will take the form shown in Fig. 2. Then the relation between the angles will be as follows:

\[ 180^\circ-(\varphi_3+\psi_3)+\varphi_3=120^\circ, \]

or

\[ 60^\circ-\varphi_3-\psi_3+\varphi_3=0 \quad \text{and} \quad \psi_3=60^\circ. \tag{5} \]

Fig. 2

Since the angle between the edges \(B_1\) and \(C_1\) in isometry is equal to \(60^\circ\), then

\[ 180^\circ-(\varphi_3+\psi_3)=60^\circ, \]

or

\[ 180^\circ-\varphi_3-60^\circ=60^\circ \quad \text{and} \quad \varphi_3=60^\circ. \tag{6} \]

From equality (3) we have

\[ \operatorname{ctg}\varphi_0=\frac{\sin(\varphi_3+\psi_3)}{\sin\varphi_3}\cdot q, \]

or

\[ \operatorname{ctg}\varphi_0=\frac{\sin120^\circ}{\sin60^\circ}\cdot 1 =\frac{\sqrt{3}\cdot 2}{2\cdot \sqrt{3}}\cdot 1=1, \]

therefore,

\[ \varphi_0=45^\circ. \tag{7} \]

From expression (4) we have

\[ \operatorname{ctg}\psi_0=\frac{\sin(\varphi_3+\psi_3)}{\sin\psi_3}\cdot r =\frac{\sin120^\circ}{\sin60^\circ}\cdot 1=1, \]

therefore,

\[ \psi_0=45^\circ. \tag{8} \]

2. Dimetry. We take the coefficients equal to \(p=v=1,\ q=\dfrac{1}{2}\).

On the basis of the preceding, the edge of the cube \(1—4=A\) in dimetry is also placed horizontally. Then the dimetric image of the cube will be turned clockwise through angle 7 and will take the form shown in Fig. 3. Then the dependence between the angles will be as follows:

\[ 180^\circ-(\varphi_3+\psi_3)+\varphi_3=131^\circ25', \]

or

\[ 48^\circ35'-(\varphi_3+\psi_3)+\varphi_3=0, \]

or

\[ 48^\circ35'-\varphi_3-\psi_3+\varphi_3=0 \]

\[ \text{and}\qquad \psi_3=48^\circ35'. \tag{9} \]

Fig. 3

Since the angle in dimetry is equal to

\[ 180^\circ-(\varphi_3+\psi_3)=48^\circ35', \]

then

\[ 180^\circ-\varphi_3-\psi_3=48^\circ35' \qquad \text{and} \qquad \varphi_3=180^\circ-\psi_3-48^\circ35', \tag{9} \]

or

\[ \varphi_3=180^\circ-48^\circ35'-48^\circ35'=82^\circ50' \qquad \text{and} \qquad \varphi_3=82^\circ50'. \tag{10} \]

On the basis of equality (3) we have

\[ \operatorname{ctg}\varphi_0=\frac{\sin(\varphi_3+\psi_3)}{\sin\varphi_3}\cdot q, \]

or

\[ \operatorname{ctg}\varphi_0= \frac{\sin(48^\circ35'+82^\circ50')}{\sin82^\circ50'}\cdot \frac{1}{2} = \frac{\sin(180^\circ-131^\circ25')}{\sin82^\circ50'\cdot 2} = \]

\[ = \frac{\sin48^\circ35'}{2\cdot \sin82^\circ50'} = \frac{0.75}{2\cdot 0.992} =0.378, \]

whence

\[ \varphi_0=69^\circ20'. \tag{11} \]

From expression (4) we have

\[ \operatorname{ctg}\psi_0= \frac{\sin(\varphi_3+\psi_3)}{\sin\psi_3}\cdot r = \frac{\sin131^\circ25'}{\sin48^\circ35'}\cdot 1 = \frac{\sin48^\circ35'}{\sin48^\circ35'}\cdot 1 =1, \]

whence

\[ \psi_0=45^\circ. \tag{12} \]

3. Frontal dimetry

We take the distortion coefficients to be equal to \(p=r=1,\quad q=\dfrac{1}{2}\).

On the basis of the preceding reasoning, the edge of the cube \(1—4=A\) will be positioned in frontal dimetry horizontally, according to Fig. 4. Then the relation between the angles of inclination of the rulers will be as follows:

\[ \varphi_0=90^\circ, \tag{13} \]

\[ 180^\circ-(\varphi_3+\psi_3)+\varphi_3=45^\circ, \]

or

\[ 180^\circ-90^\circ-\psi_3+90^\circ=45^\circ, \]

whence

\[ \psi_3=45^\circ. \tag{14} \]

Fig. 4

On the basis of expression (3) we have

\[ \operatorname{ctg}\varphi_0= \frac{\sin(\varphi_3+\psi_3)}{\sin\varphi_3}\cdot q = \frac{\sin135^\circ}{\sin90^\circ}\cdot\frac{1}{2} = \frac{\sqrt{2}\cdot1}{2\cdot1\cdot2} =0.3535, \]

whence

\[ \varphi_0=70^\circ35'. \tag{15} \]

From expression (4) we have

\[ \operatorname{ctg}\psi_0= \frac{\sin(\varphi_3+\psi_3)}{\sin\psi_3}\cdot r = \frac{\sin135^\circ}{\sin45^\circ}\cdot1 = \frac{\cos45^\circ}{\sin45^\circ}\cdot1 = \frac{\sqrt{2}}{2}\cdot\frac{2}{\sqrt{2}} =1, \]

whence

\[ \psi_0=45^\circ. \tag{16} \]

Thus the values of the angles of inclination of the rulers to the sliders can be summarized in the following table:

On the basis of the theoretical reasoning set forth above, a construction of the simplified axonograph has been created, shown in Fig. 5.

The axonograph consists of two horizontal rulers \(I\) and \(II\), which can move horizontally along guide lines \(M\) and \(N\), installed on the drawing board in the appropriate place by means of needles or points at their ends. To the sliders, on screws 1, 2, 3, 4, are attached rulers \(III\), \(IV\), \(V\), \(VI\), which, rotating on the screws, can be set at the required angles \(\varphi_0\), \(\psi_0\), \(\varphi_3\), \(\psi_3\) with respect to sliders \(I\) and \(II\). Screws 1, 2, 3, 4 can, in turn, move horizontally in the slots of sliders \(M\) and \(N\), and thereby allow different mutual positions of rulers \(III\) and \(V\), \(IV\) and \(VI\), depending on where the axonometric image of the object is to be constructed. Figure 5 shows the setting of the instrument for constructing a rectangular

dimetry of the nut. Rulers III and IV are set, respectively, on the horizontal and vertical projections of point 8 \((8, 8')\) of the nut edge 7—8; then at the intersection of rulers V and VI the dimetric projection of this point 8 is marked, and it is even possible to draw in dimetry the direction of the entire edge of the nut 7—8 along the edge of ruler V.

Fig. 5

With the aid of the device shown in Fig. 5, an isometry of a cube has been constructed—Fig. 6. On the horizontal positions of sliders I and II, the positions of rulers III, IV, V, VI for constructing the isometric images of the corresponding vertices of the cube are marked with numbers. The inclined rulers are arranged with respect to the sliders at the angles

\[ \varphi_0 = 45^\circ, \qquad \psi_0 = 45^\circ, \]

\[ \varphi_3 = 60^\circ, \qquad \psi_3 = 60^\circ. \]

In Fig. 7 the dimetric projection of the same cube is shown. In this case the angles of inclination of rulers III, IV, V, VI are taken as equal to

\[ \varphi_0 = 69^\circ 20', \qquad \psi_0 = 45^\circ, \qquad \varphi_3 = 82^\circ 50', \qquad \psi_3 = 48^\circ 35'. \]

Fig. 6

In Fig. 8 the construction of the frontal dimetry of a cube is shown. The angles of inclination of rulers III, IV, V, VI are taken as equal to

\[ \varphi_0 = 70^\circ 35', \qquad \psi_0 = 45^\circ, \qquad \varphi_3 = 90^\circ, \qquad \psi_3 = 45^\circ. \]

Fig. 7

Fig. 8

Fig. 9

If the inclination of rulers III, IV, V, VI is changed to the left side of the vertical line of the drawing, and a rightward inclination is taken as positive and a leftward one as negative, then such a change of inclination will be equivalent to changing the direction of projection onto the axonometric plane from the right side to the left. Such a change of angles is shown in Fig. 9, where the dimetry of a cube is constructed.

If one of the angles, for example, \(\psi_0\), is made negative, but equal to \(45^\circ\), then the cube appears as if overturned (Fig. 10). The sight rays will be directed along the arrows shown on lines III and IV. The cube will be viewed from below, from vertex 7.

Fig. 10

If the angle \(\varphi_0\) is changed to a negative one, but still equal to \(\varphi_0 = 69^\circ 20'\), then the cube will be viewed in the direction of a sight ray directed from vertex 4 upward from below.

If only both inclinations of rulers III and IV are changed, i.e., \(\varphi_0\) and \(\psi_0\), to negative ones, but the values are left as \(\varphi_0 = 69^\circ 20'\) and \(\psi_0 = 45^\circ\), then, consequently, with the directions of the sight rays from the vertex from left to right, the cube will be viewed in an overturned position.

If the inclination angles \(\varphi_0\) and \(\psi_0\) are left positive and equal to \(\varphi_0 = 69^\circ 20'\), \(\psi_0 = 45^\circ\), while the angles \(\varphi_3\) and \(\psi_3\) are made negative but equal in magnitude to \(\varphi_3 = 82^\circ 50'\) and \(\psi_3 = 48^\circ 35'\), then the cube will be viewed from vertex 3, from left to right.

If the angles \(\varphi_3 = 82^\circ 50'\) and \(\psi_3 = 48^\circ 35'\) are left negative, the angle \(\varphi_0 = 69^\circ 20'\) is also given a negative direction, and the angle \(\psi_0 = 45^\circ\) is left positive, then the cube will be viewed in an overturned form from the rear, from left to right.

If the angles \(\varphi_3 = 82^\circ 50'\) and \(\psi_3 = 48^\circ 35'\) are left negative, the angle \(\varphi_0 = 69^\circ 20'\) is left positive, and the angle \(\psi_0 = 45^\circ\) is changed to negative, then the image of the cube will be represented as a view from vertex 1, from below upward, from left to right.

It is also possible to interchange the positions of rulers III and IV, V and VI, and with them the inclination angles \(\varphi_0\) and \(\psi_0\), \(\varphi_3\) and \(\psi_3\). If ruler III is transferred to the upper slider II at the angle \(\varphi_0 = 69^\circ 20'\), and ruler IV is moved to slider I at the angle \(\psi_0 = 45^\circ\), and the same is done with rulers V and VI, arranging them at the angles \(\varphi_3 = 82^\circ 50'\) and \(\varphi_3 = 48^\circ 35'\) to the sliders, then the cube will be projected from the side of vertex 5, from left to right, in a position inclined toward the observer.

Unfortunately, the limited length of the article does not permit us to include drawings of the listed variants of the images.

The same changes in the position of the rulers may also be made for frontal dimetry. However, the examples given with the cube are sufficient to indicate the universality of the proposed construction, which makes it possible to construct mechanically axonometric images of an object from different viewpoints, and moreover with distortion coefficients recommended by GOST 3453—59 for isometry, dimetry, and frontal dimetry: \(1:1:1\), \(1:\tfrac{1}{2}:1\), \(1:\tfrac{1}{2}:1\).

As an application of the axonograph, Fig. 11 presents an image of a sphere with a three-sided cutout. To produce such an image, ...

the points belonging to the lines of the cutout, 15 in number, are first marked on the horizontal and vertical projections; then the rulers of the instrument are set at the angles \(\varphi_0=69^\circ20'\), \(\psi_0=45^\circ\), \(\varphi_3=82^\circ50'\), \(\psi_3=48^\circ35'\) (Fig. 5), and the axonometric projections of all the points are found with the aid of the instrument. The construction of such an image took 15 minutes, including inking.

Fig. 11

It should be noted that the axis \(O_1X_1\) in dimetry turns out to be rotated clockwise by \(7^\circ\), which, of course, does not change the quality of the image obtained. To preserve the customary position of the axis \(O_1X_1\), inclined to the horizontal line at an angle of \(7^\circ\) downward, one may recommend constructing the dimetry (or in general any axonometry) on a separate drawing sheet, pinned to the drawing board next to the given orthogonal projections and rotated clockwise by \(7^\circ\) with its left side. With this position of the sheet, the axis \(O_1X_1\) will assume the usual position relative to its lower side.

Figure 12 shows the construction in dimetry of a pyramid with a cutout, and in Fig. 13 the angles for rulers III and IV are taken as negative. The construction of such a figure required only 10 minutes with the error-free construction of the image. In Fig. 14 the construction of the dimetric axonometry of a technical part is carried out from its given projections. The execution of Fig. 14 required 25 minutes. With the ordinary construction of such an image, considerably more time would have been spent, with possible and inevitable errors in complex figures, especially in constructing the lines of intersection of curved surfaces.

Let us point out one more feature of the proposed construction, namely the possibility of constructing the vertical and horizontal projections of an object from its axonometric image. We shall show this construction in Fig. 15.

Suppose that an image of a cube is given in rectangular dimetry, and that, in order to construct its orthogonal projections, we proceed as follows. We place the instrument on the drawing, as described earlier; rulers III, IV, V, VI are set at the required angles \(\varphi_0\), \(\psi_0\), \(\varphi_3\), \(\psi_3\) to sliders I and II, as in constructing dimetry. Then rulers V and VI are placed on the dimetric projection of some point, for example, 1, and near the edge of ruler III we mark point 1 at any place along it; and along the edge of ruler IV, on the vertical line from 1, we mark the vertical projection of point \(1'\). Ruler VI is then moved to the poin-

Fig. 12

Fig. 13

Fig. 14

to 5, and on the same vertical line where point 1 was located, we mark point \(5'\) along the edge of the ruler. In this way all points of the cube are traversed and their horizontal and vertical projections are determined, which are connected by straight lines. Of course, the mutual position of the horizontal and vertical projections of points on the connecting lines can be marked only on graph paper, or on a glass table with such a millimeter grid illuminated from below, or else if there is also a vertical thin metal ruler that can move under rulers III and IV only in the horizontal direction, while remaining parallel to itself.

Fig. 15

Operation of the device is extremely simple and is easily mastered by persons not even familiar with the theory of axonometry. Such a device may be recommended to students when carrying out work on constructing axonometry, as is in fact done at the Tomsk Polytechnic Institute.

The following procedure for using the device may be recommended:

1. Pin the drawing (in two projections) to the drawing board in such a way that the \(OX\) axis is parallel to the lower edge of the board.

2. Pin the sheet of paper for constructing the axonometric projection so that the left edge of the sheet is turned clockwise by \(30^\circ\) for isometry, and by \(7^\circ\) for dimetry. Then the axonometric image of the object will assume the usual normal position with respect to the edges of the sheet.

3. Set up the device, placing the rulers at the angles required for the sliders; this is easily accomplished by constructing these angles in the field of the drawing with the aid of a protractor, and arranging the inclinations of the rulers according to them. The distances between rulers III and V, IV and VI are chosen depending on the position of the axonometric image.

4. On the paper for axonometry, first construct the axonometric axes \(O_1X_1\), \(O_1Y_1\), \(O_1Z_1\), and check the position of all the principal points of the future image—the extreme left, extreme right, lowest, and highest—to make sure that the axonometry can be fitted onto the given sheet of paper.

5. After this, one may proceed to construct the axonometry of the individual points of the object with the device, of course having first marked a number of points in the given projections.

CORRECTIONS AND NOTICED MISPRINTS