P. A. Maslennikov.

Orthogonal Axonometric Epure.

For the solution of various problems connected with the construction of images of spatial forms by the orthogonal method of projection, it is known to be sufficient to have only two images of the spatial forms on two mutually perpendicular planes of projection, for example, the vertical and horizontal planes of projection.

When constructing drawings of various spatial forms, however, one usually does not limit oneself to only two images on two planes of projection, but resorts, depending on the complexity of the body being represented, to constructing images on three, four, etc., planes of projection representing the faces of a cube. And nevertheless, from the existing orthogonal images of a body of complex form, it is often very difficult—especially for a person with little training—to recreate in the imagination the spatial appearance of the projected body; therefore, especially in recent times, in addition to the orthogonal images of the body on the faces of the cube, so-called axonometric (orthogonal or oblique) images are sometimes appended in order to facilitate the reading of the drawing.

The existing construction of an axonometric image of a body, separate from the orthogonal constructions of the drawing, owing to the changes of scale necessary for such images (especially orthogonal axonometric ones), requires of the person preparing the axonometric image of the projected body the necessary knowledge and skills in choosing scales and the arrangement of axes for various forms, and takes a very great deal of time to prepare. Moreover, the separate construction of an axonometric image diminishes its value because it is not always possible quickly to establish which line of the drawing corresponds to a given line of the axonometric image, and conversely.

The absence of a visible connection between the orthogonal projections of the drawing and the axonometric image compels one to regard the latter as an expensive supplement to the principal drawing, and therefore the limited prevalence of axonometric images is understandable.

In the present work I propose a method for the simultaneous construction both of the orthogonal projections of a drawing on the two principal planes of projection (vertical and horizontal) and of an axonometric image, by creating an epure that preserves the integrity of all the images and the unity of their constructions.

The essence of the proposed epure, which hereafter I shall call an orthogonal axonometric epure, consists in introducing into the space of the right dihedral angle formed by the principal planes of projection (vertical and horizontal) a third plane, inclined to both principal planes of projection, forming with the first two such angles as would ensure, under orthogonal projection of the form onto the third plane, sufficient visual clarity of the resulting axonometric image on it.

Without dwelling on the method of development of these three planes and on the construction of the epure of a point, which was set forth in sufficient detail in my work

Substituting the value of \(f_{pi}\) into equation (3), we obtain:

\[ U=\frac{1}{2}\sum_{i=0}^{i=n}\frac{f_{pi}^{2}}{\alpha_{pi}}. \]

And, consequently:

\[ \frac{\partial U}{\partial f_{pk}}=\frac{f_{pk}}{\alpha_{pk}}=P_k. \]

The graph of the second variant of the proof of Castigliano’s theorem (Fig. 4) leads to an easy proof of Betti’s theorem in its most general form.

The condition of equality of the sum of the areas of the displacement diagrams, when the system is loaded in one way and in the other way, is expressed analytically as follows:

\[ \frac{1}{2}\sum_{0}^{n} P_i f_{ii} +P_0\sum_{1}^{n} f_{0i} +P_1\sum_{2}^{n} f_{1i} +P_2\sum_{3}^{n} f_{2i} +\cdots = \]

\[ = \frac{1}{2}\left[ P_0\sum_{0}^{n} f_{0i} +P_1\sum_{0}^{n} f_{1i} +P_2\sum_{0}^{n} f_{2i} +\cdots \right]. \]

Or, after simplification, owing to the mutual cancellation of individual terms of the equality:

\[ P_0\sum_{1}^{n} f_{0i} +P_1\sum_{2}^{n} f_{1i} +P_2\sum_{3}^{n} f_{2i} +\cdots +P_{n-1}f_{(n-1)n} = \]

\[ = P_n\sum_{0}^{n-1} f_{ni} +P_{n-1}\sum_{0}^{n-2} f_{(n-1)i} +P_{n-2}\sum_{0}^{n-3} f_{(n-2)i} +\cdots +P_1 f_{10}. \]

The last equality can already be written in an entirely compact form:

\[ \sum_{k=0}^{k=n-1} P_k \sum_{i=k+1}^{i=n} f_{ki} = \sum_{k=1}^{k=n} P_k \sum_{i=0}^{i=k-1} f_{ki}. \]

Betti’s theorem, obtained in such an analytic formulation, could be stated in the following way:

If all the loads deforming a system are arranged in an arbitrary numerical sequence, then the sum of the products of the loads by their displacements from the action of all the loads lying from the beginning of the sequence up to them is equal to the sum of the products of the loads by their displacements from the action of all the loads lying from them to the end of the sequence.

Giving the value:

\[ k=0 \]

we obtain:

\[ P_0 f_{01}=P_1 f_{10}, \]

i.e., the special case first shown by Maxwell: every pair of loads in an elastically deformed system is related in such a way that equality holds between the products of each load by its displacement from the action of the other load.

b) We shall express the values of the angles between the axonometric axes through the distortion scales and the angles \(\alpha\) and \(\beta\), for which:

1. From the oblique triangles \(bO_1c\), \(aO_1c\), and \(aO_1b\) we determine the values of the angles \(\omega_x\), \(\omega_y\), and \(\omega_z\) through the segments of the axonometric coordinates \(x_1\), \(y_1\), and \(z_1\), and through the segments of the straight lines \(bc\), \(ac\), and \(ab\).

2. From the right triangles \(bOc\), \(aOc\), and \(aOb\) we express, through the segments of the spatial coordinates \(x\), \(y\), and \(z\), the values of the segments of the straight lines \(bc\), \(ac\), and \(ab\).

3. We express the magnitudes of the segments \(bc\), \(ac\), and \(ab\), and the magnitudes of the segments along the coordinate axes, in the found values of the angles \(\omega_x\), \(\omega_y\), and \(\omega_z\), through the distortion scales \(m_x\), \(m_y\), and \(m_z\), and the angles \(\alpha\) and \(\beta\).

c) Derivation of the distortion scales and of the values of the angles between the axonometric axes.

From the right triangles \(cOk\) and \(Oka\) (Fig. 1) we find:

Fig. 1.

\[ tg\,\omega_h = tg \angle Okc = \frac{Oc}{Ok} = \frac{z}{Ok} \qquad \ldots \ldots \ldots (a) \]

\[ Ok = Oa \operatorname{Sin} \angle Oak = x \operatorname{Sin}(180^\circ-\alpha)=x\operatorname{Sin}\alpha \qquad \ldots \ldots \ldots (b) \]

Substituting from \((b)\) the value of \(Ok\) into formula \((a)\), we obtain:

\[ tg\,\omega_h=\frac{z}{x}\cdot \frac{1}{\operatorname{Sin}\alpha} \qquad \ldots \ldots \ldots (1) \]

From the right triangle \(kO_1O\) we find:

\[ OO_1 = Ok \operatorname{Sin}\omega_h = x\operatorname{Sin}\alpha\, \frac{tg\,\omega_h}{\sqrt{1+tg^2\omega_h}} \]

or, after substituting the value of \(tg\,\omega_h\) from formula (1),

\[ OO_1=\frac{x\cdot z\cdot \operatorname{Sin}\alpha} {\sqrt{x^2\operatorname{Sin}^2\alpha+z^2}} \qquad \ldots \ldots \ldots (2) \]

We shall transform the obtained value of the length of the perpendicular \(OO_1\) in such a way that its expression contains only one or another segment of the rectangular spatial coordinates, for which we express through the angles

“Projection onto three planes,” I shall turn to the questions of determining the scales of distortion obtained on the third plane along the coordinate axes and the angles between the axonometric axes, the classification of the images obtained, and the choice of the type of axonometric image.

I. Determination of the scales of distortion and of the angles between the axonometric axes.

A. Explanations for the drawings and adopted notation (drawings 1, 2, and 3).

$H$ and $V$ — the horizontal and vertical planes of projection.

$O$ — the origin of rectangular spatial coordinates.

$OX$, $OY$, and $OZ$ — the axes of rectangular spatial coordinates.

$aM_h$ and $aM_v$ — the horizontal and vertical traces of the orthoaxonometric plane $M$ on the planes of projection $H$ and $V$ 1).

$Oa=x$, $Ob=y$, and $Oc=z$ — the segments of rectangular coordinates cut off by the orthoaxonometric plane $M$ on the coordinate axes.

$OO_1$ — the perpendicular to the orthoaxonometric plane $M$, drawn from the origin of coordinates $O$.

$O_1$ — the origin of axonometric coordinates.

$O_1a=x_1$, $O_1b=y_1$, and $O_1c=z_1$ — the segments of axonometric coordinates on the plane $M$.

$\alpha$ — the angle formed by the positive direction of the $X$ axis with the horizontal trace $aM_h$ of the plane $M$ on the plane $H$. It is measured clockwise.

$\beta$ — the angle formed by the positive direction of the $X$ axis with the vertical trace $aM_v$ of the plane $M$ on the plane $V$. It is measured clockwise.

$bl$ — the line of intersection of the orthoaxonometric plane $M$ with the auxiliary plane $bOl$ perpendicular to the trace $aM_v$.

$ck$ — the line of intersection of the orthoaxonometric plane $M$ with the auxiliary plane $cOk$ perpendicular to the trace $aM_h$.

$\omega_h$ — the angle formed by the orthoaxonometric plane $M$ with the horizontal plane of projection $H$.

$\omega_x$, $\omega_y$, and $\omega_z$ — the angles formed by the axonometric coordinate axes (drawing 3).

$m_x=\dfrac{x_1}{x}$, $m_y=\dfrac{y_1}{y}$, and $m_z=\dfrac{z_1}{z}$ — the scales of distortion along the coordinate axes.

B. Scheme for determining the scales of distortion and the angles between the axonometric axes.

a) Express the scales of distortion along the coordinate axes through the given angles $\alpha$ and $\beta$, for which purpose (drawings 1, 2, and 3)

1. From the right triangle $cOk$ we determine the magnitude of the angle $\omega_h$.

2. From the right triangle $kO_1O$ we determine the length of the perpendicular $OO_1$.

3. By solving three right triangles $aO_1O$, $bO_1O$, and $cO_1O$, we express the values of the segments of axonometric coordinates $x_1$, $y_1$, and $z_1$ in terms of the segments of rectangular coordinates $x$, $y$, and $z$ and the angles $\alpha$ and $\beta$.

4. We determine the scales of coordinate distortions by dividing the segments of axonometric coordinates by the corresponding segments of rectangular spatial coordinates.

1) The orthoaxonometric plane $M$ in the work “Projection onto three planes” is called the principal plane of the figure being projected.

coordinates of the axes, through the angles \(\alpha\) and \(\beta\), for the case of orthogonal projection:

\[ m_x^2=\frac{\tg^2\alpha+\tg^2\beta}{\tg^2\alpha+\tg^2\beta+\tg^2\alpha\,\tg^2\beta} \qquad . . . . . (6) \]

\[ m_y^2=\frac{\tg^2\alpha+\tg^2\alpha\,\tg^2\beta}{\tg^2\alpha+\tg^2\beta+\tg^2\alpha\,\tg^2\beta} \qquad . . . . . (7) \]

\[ m_z^2=\frac{\tg^2\beta+\tg^2\alpha\,\tg^2\beta}{\tg^2\alpha+\tg^2\beta+\tg^2\alpha\,\tg^2\beta} \qquad . . . . . (8) \]

The formulas obtained, (6), (7), and (8), for the scales of distortion along the coordinate axes, are the general formulas for the case of orthogonal projection.

By solving the oblique triangles \(bOc\), \(aOc\), and \(aOb\) (Fig. 3), we express the values of the angles between the axonometric axes through the segments of the axonometric coordinates and the segments of the straight lines \(cb\), \(ac\), and \(ab\).

Fig. 3.

\[ \left. \begin{aligned} \Cos\omega_x&=\frac{y_1^2+z_1^2-cb^2}{2y_1z_1}\\[4pt] \Cos\omega_y&=\frac{x_1^2+z_1^2-ac^2}{2x_1z_1}\\[4pt] \Cos\omega_z&=\frac{x_1^2+y_1^2-ab^2}{2x_1y_1} \end{aligned} \right\} \qquad . . (8) \]

But from the right triangles \(bOc\), \(aOc\), and \(aOb\), we obtain:

\[ cb^2=y^2+z^2, \]

\[ ac^2=x^2+z^2 \ \text{and} \]

\[ ab^2=x^2+y^2 \]

After substituting the values \(cb^2\), \(ac^2\), and \(ab^2\) into formulas (8), we find:

\[ \Cos\omega_x=\frac{y_1^2+z_1^2-y^2-z^2}{2y_1z_1} \]

\[ \Cos\omega_y=\frac{x_1^2+z_1^2-x^2-z^2}{2x_1z_1} \]

\[ \Cos\omega_z=\frac{x_1^2+y_1^2-x^2-y^2}{2x_1y_1} \]

or, after expressing the values of the coordinate segments through the scales of distortion along the coordinate axes and the angles \(\alpha\) and \(\beta\), we obtain the final expression for the angles between the axonometric axes in the form:

\[ \Cos\omega_x= -\frac{(m_y^2-1)\tg^2\alpha+(m_z^2-1)\tg^2\beta}{2m_y m_z\tg\alpha\,\tg\beta} \qquad . . . . . (9) \]

\[ \Cos\omega_y= -\frac{(m_x^2-1)+(m_z^2-1)\tg^2\beta}{2m_xm_z\tg\beta} \qquad . . . . . (10) \]

the connection between the segments of rectangular spatial coordinates, $\alpha$ and $\beta$, by solving two right triangles $aOb$ and $aOc$.

Solving the named triangles, we obtain:

\[ \frac{Ob}{Oa}=\frac{y}{x}=\operatorname{tg}(180^\circ-\alpha)=\operatorname{tg}\alpha \]

whence

\[ x=-y\operatorname{Cotg}\alpha \ . . . . . \ (c) \]

and

\[ \frac{Oc}{Oa}=\frac{z}{x}=\operatorname{tg}\angle caO=\operatorname{tg}(\beta-180^\circ)=\operatorname{tg}\beta \]

or:

\[ z=x\operatorname{tg}\beta \ . . . . . \ (d) \]

\[ x=z\operatorname{Cotg}\beta \ . . . . . \ (e) \]

and from equalities $(c)$ and $(d)$:

\[ z=-y.\operatorname{Cotg}\alpha\,\operatorname{tg}\beta \ . . . . . . . . . . . . (f) \]

Fig. 2.

Substituting the value of $z$ from equality $(d)$, the values of $x$ and $z$ from equalities $(c)$ and $(f)$, and the value of $x$ from equality $(e)$ into formula (2), we obtain:

\[ OO_1=x\frac{\operatorname{Sin}\alpha\,\operatorname{tg}\beta} {\sqrt{\operatorname{Sin}^2\alpha+\operatorname{tg}^2\beta}} \ . . . . . . . . . . (2_x) \]

\[ OO_1=y\frac{\operatorname{Cos}\alpha\,\operatorname{tg}\beta} {\sqrt{\operatorname{Sin}^2\alpha+\operatorname{tg}^2\beta}} \ . . . . . . . . . . (2_y) \]

\[ OO_1=z\frac{\operatorname{Sin}\alpha} {\sqrt{\operatorname{Sin}^2\alpha+\operatorname{tg}^2\beta}} \ . . . . . . . . . . (2_z) \]

From the right triangles $aO_1O$, $bO_1O$ and $cO_1O$ we find the values of the segments of the axonometric coordinates:

\[ x_1^2=(O_1a)^2=(Oa)^2-(OO_1)^2=x^2-(OO_1)^2 \]

\[ y_1^2=(O_1b)^2=(Ob)^2-(OO_1)^2=y^2-(OO_1)^2 \]

\[ z_1^2=(O_1c)^2=(Oc)^2-(OO_1)^2=z^2-(OO_1)^2 \]

or, after substituting from formulas $(2_x)$, $(2_y)$ and $(2_z)$ the value of $OO_1$ and simple transformations:

\[ x_1^2=x^2\cdot \frac{\operatorname{tg}^2\alpha+\operatorname{tg}^2\beta} {\operatorname{tg}^2\alpha+\operatorname{tg}^2\beta+\operatorname{tg}^2\alpha\,\operatorname{tg}^2\beta} \ . . . . . (3) \]

\[ y_1^2=y^2\cdot \frac{\operatorname{tg}^2\alpha+\operatorname{tg}^2\alpha\,\operatorname{tg}^2\beta} {\operatorname{tg}^2\alpha+\operatorname{tg}^2\beta+\operatorname{tg}^2\alpha\,\operatorname{tg}^2\beta} \ . . . . . (4) \]

\[ z_1^2=z^2\cdot \frac{\operatorname{tg}^2\beta+\operatorname{tg}^2\alpha\,\operatorname{tg}^2\beta} {\operatorname{tg}^2\alpha+\operatorname{tg}^2\beta+\operatorname{tg}^2\alpha\,\operatorname{tg}^2\beta} \ . . . . . (5) \]

Dividing the obtained expressions for $x_1^2$, $y_1^2$ and $z_1^2$ respectively by $x^2$, $y^2$ and $z^2$, we obtain the following expressions for the distortion scales along

the angle \(\alpha\) must be a multiple of \(45^\circ\). Substituting the obtained value \(\tg \alpha\) into the general formulas, we find for this case:

\[ \left. \begin{aligned} m_x^2&=\frac{1+\tg^2\beta}{1+2\tg^2\beta}\\ m_y^2&=\frac{1+\tg^2\beta}{1+2\tg^2\beta}\\ m_z^2&=\frac{2\tg^2\beta}{1+\tg^2\beta} \end{aligned} \right\} \qquad \ldots\ldots\ldots (12) \]

Second group of the class of dimetric projections (Fig. 6):

\[ m_x=m_z\ne m_y \]

Equating eqs. (6) and (8), we find: \(\tg^2\beta=1\), i.e., for the second group of the class of dimetric projections, with arbitrary angle \(\alpha\), the value of angle \(\beta\) must be a multiple of \(45^\circ\). Substituting the obtained value \(\tg \beta\) into the general formulas, for this group we find:

\[ \left. \begin{aligned} m_x^2&=\frac{1+\tg^2\alpha}{1+2\tg^2\alpha}\\ m_y^2&=\frac{2\tg^2\alpha}{1+2\tg^2\alpha}\\ m_z^2&=\frac{1+\tg^2\alpha}{1+2\tg^2\alpha} \end{aligned} \right\} \qquad \ldots (13) \]

Fig. 6.

Third group of the class of dimetric projections (Fig. 7):

\[ m_y=m_z\ne m_x \]

Equating eqs. (7) and (8), we find: \(\tg^2\alpha=\tg^2\beta\), i.e., for the third group of the class of dimetric projections it must be \(\beta=\alpha\), or \(\beta=180^\circ\pm\alpha\). Substituting the accepted value of angle \(\alpha\) or \(\beta\) into the general formulas, for this group we find:

\[ \left. \begin{aligned} m_x^2&=\frac{2}{2+\tg^2\alpha}\\ m_y^2&=\frac{1+\tg^2\alpha}{2+\tg^2\alpha}\\ m_z^2&=\frac{1+\tg^2\alpha}{2+\tg^2\alpha} \end{aligned} \right\} \qquad \ldots (14) \]

Fig. 7.

\[ \operatorname{Cos}\omega_z = - \frac{(m_x^2-1)+(m_y^2-1)\operatorname{tg}^2\alpha} {2m_xm_y\operatorname{tg}\alpha} \qquad \ldots\ldots (11) \]

The last formulas will also be the general formulas for determining the angles between the axonometric axes under rectangular projection.

II. Classification of the Image.

It is not difficult to see that, for arbitrary values of the angles \(\alpha\) and \(\beta\), the general formulas (6), (7), and (8) give different values of the distortion scales along the coordinate axes, that is, one obtains:

\[ m_x \ne m_y \ne m_z \]

and such inequalities, as is known, define the so-called trimetric projections; consequently, the general formulas for the distortion scales along the coordinate axes (6), (7), and (8) are at the same time formulas for the distortion scales along the coordinate axes for the class of trimetric projections (Fig. 4).

Fig. 4.

To obtain formulas expressing the distortion scales along the coordinate axes for the class of dimetric projections, for which two scales are equal to each other and not equal to the third, by the joint solution of any two equations (6), (7), and (8) and by substituting the obtained values of the angles \(\alpha\) and \(\beta\) into the third, we find the following expressions for the distortion scales along the coordinate axes for three groups of the class of dimetric orthogonal projections.

1st group of the class of dimetric projections (Fig. 5):

\[ m_x = m_y \ne m_z \]

Fig. 5.

Equating eqs. (6) and (7), we find: \(\operatorname{tg}^2\alpha = 1\), i.e. for the first group of the class of dimetric projections, with arbitrary angle \(\beta\), the value

III. Choice of the Type of Axonometric Representation

In order to choose the type of axonometric representation—or, more precisely, to establish for a given epure the magnitudes of the angles $\alpha$ and $\beta$ that determine the position of the orthoaxonometric plane relative to the principal planes of projection—one should proceed chiefly from the condition of sufficient clarity of the image obtained on the orthoaxonometric plane; the necessary condition of uniformity or simplicity of the scales of distortion along the axonometric axes is irrelevant in the present case, since the construction of an image in an orthoaxonometric epure does not depend on the magnitudes of the scales of distortion.

From consideration of the axonometric representations of a cube in the accompanying drawings 4, 5, 6, 7, and 8, it is evident that the least clear among them are the images of the isometric class (drawing 8) and of the dimetric class: group 1 (drawing 5) and group 2 (drawing 6), that is, precisely those which are most frequently used, owing to the simplicity of their construction, in constructions separate from the orthogonal projections of the drawing. The remaining types of axonometric representations—the class of trimetric representations (drawing 4) and group 3 of the dimetric class (drawing 7)—as is not difficult to see from the accompanying drawings, possess sufficient clarity and therefore should be recommended for constructions of an orthoaxonometric epure; moreover, the latter type (group 3 of the dimetric class), for which both traces of the orthoaxonometric plane form a single straight line, will be the most acceptable, as possessing greater simplicity of construction.

or:

\[ \left. \begin{aligned} m_x^2 &= \frac{2}{2+\operatorname{tg}^2\beta} \\[1.5em] m_y^2 &= \frac{1+\operatorname{tg}^2\beta}{2+\operatorname{tg}^2\beta} \\[1.5em] m_z^2 &= \frac{1+\operatorname{tg}^2\beta}{2+\operatorname{tg}^2\beta} \end{aligned} \right\} \qquad \ldots\ldots\ldots (15) \]

Making in formulas (6), (7), and (8) all scales equal, that is, putting \(m_x=m_y=m_z\), we obtain the so-called class of isometric projections

Fig. 8.

(Fig. 8), the values of the angles \(\alpha\) and \(\beta\) for which are determined from the equalities \(\operatorname{tg}^2\alpha=1\) and \(\operatorname{tg}^2\beta=1\), that is, both \(\alpha\) and \(\beta\) must be multiples of \(45^\circ\), and the distortion scales along the coordinate axes will be equal:

\[ \left. \begin{aligned} m_x^2 &= \frac{2}{3} \\[1.5em] m_y^2 &= \frac{2}{3} \\[1.5em] m_z^2 &= \frac{2}{3} \end{aligned} \right\} \qquad \ldots\ldots\ldots (16) \]