Methodology of Parallel Projection and Its Application to Mapping the Exploitation of Thick and Steeply Dipping Deposits

P. A. Maslennikov

Introduction

In mining a suite of gently dipping seams, or even a single steeply dipping seam (or vein), a plan of the extraction of the mineral deposit, constructed by the method of orthogonal projection onto one horizontal plane of projection, usually does not satisfy all the requirements of deposit exploitation; therefore it is very often necessary to resort to constructing auxiliary orthogonal representations of the workings on a vertical plane or on a plane parallel to the plane of the seam being represented. Even so, from the representations of the workings obtained by orthogonal projection, since they do not possess good visual clarity, it is very difficult to reconstruct in the imagination the spatial appearance of the workings shown; it is difficult to solve problems connected with determining directions, and still more difficult to design new workings.

All that has been said above suggests the introduction of another method of projecting mineral workings—namely, one which, while possessing sufficient ease of measurement, would at the same time satisfy the condition of the necessary visual clarity; that is, would provide such representations as would make it possible to see, on a single plane of projection, all three dimensions of the form being represented at the same time.

Among the methods of projection that satisfy the stated requirements, I shall dwell on the so-called oblique-angle axonometric projections, obtained as a result of applying parallel projection of spatial forms onto a plane that makes an angle other than a right angle with the direction of projection, and which at the same time are also the simplest to construct.

Derivation of the General Formulas of Parallel Projection onto a Plane

Let three mutually perpendicular segments \(OA\), \(OB\), and \(OC\), which are the rectangular coordinates \(x\), \(y\), and \(z\) of a certain point and which make, with the perpendicular \(OO_2\) to the plane of projection \(M\), the angles \(\alpha\), \(\beta\), \(\gamma\), be projected onto the plane \(M\) parallel to a given direction \(S\), making with the perpendicular to the plane the angle \(\varphi\), and with the straight lines \(OA\), \(OB\), and \(OC\) the angles \(\alpha_1\), \(\beta_1\), and \(\gamma_1\) (Fig. 1).

As is known from the foundations of parallel projection, on the plane of projection \(M\) the segments \(OA\), \(OB\), and \(OC\) must likewise be represented by segments of straight lines, in the general case not equal to their true length.

Let the lengths of the images of the coordinate segments become equal to
\(O_1A=x_1,\ O_1B=y_1\), and \(O_1C=z_1\) (Fig. 1). Denoting the ratios \(\dfrac{x_1}{x}\), \(\dfrac{y_1}{y}\), and \(\dfrac{z_1}{z}\), respectively, by \(m_x,\ m_y\), and \(m_z\), and calling them the scales of distortion along the coordinate axes, let us express the values of the latter in terms of the angles \(\alpha,\ \beta,\ \gamma,\ \alpha_1,\ \beta_1,\ \gamma_1\), and \(\varphi\).

From the right triangles \(OO_2A,\ OO_2B,\ OO_2C\), and \(OO_2O_1\), we find:

\[ OO_2 = x\cos\alpha = y\cos\beta = z\cos\gamma \tag{a} \]

and

\[ OO_1=\frac{OO_2}{\cos\varphi} = x\frac{\cos\alpha}{\cos\varphi} = y\frac{\cos\beta}{\cos\varphi} = z\frac{\cos\gamma}{\cos\varphi}. \tag{b} \]

Fig. 1

Solving the oblique triangles \(OO_1A,\ OO_1B\), and \(OO_1C\), we shall have:

\[ x_1^2=x^2+OO_1^2-2x\cdot OO_1\cdot\cos\alpha_1 \]

\[ y_1^2=y^2+OO_1^2-2y\cdot OO_1\cdot\cos\beta_1 \]

\[ z_1^2=z^2+OO_1^2-2z\cdot OO_1\cdot\cos\gamma_1 \]

or, after substituting the value of \(OO_1\) from (b):

\[ x_1^2=x^2+x^2\frac{\cos^2\alpha}{\cos^2\varphi} -2x^2\frac{\cos\alpha}{\cos\varphi}\cdot\cos\alpha_1, \]

\[ y_1^2=y^2+y^2\frac{\cos^2\beta}{\cos^2\varphi} -2y^2\frac{\cos\beta}{\cos\varphi}\cdot\cos\beta_1, \]

\[ z_1^2=z^2+z^2\frac{\cos^2\gamma}{\cos^2\varphi} -2z^2\frac{\cos\gamma}{\cos\varphi}\cdot\cos\gamma_1. \]

Dividing the equations obtained, respectively, by \(x^2,\ y^2\), and \(z^2\), after simple transformations we obtain the following general formulas expressing the scales of distortion along the coordinate axes:

\[ \left. \begin{aligned} m_x^2 &= \frac{1}{\cos^2\varphi} \left[\cos^2\varphi+\cos^2\alpha-2\cos\varphi\cos\alpha\cos\alpha_1\right], \\[6pt] m_y^2 &= \frac{1}{\cos^2\varphi} \left[\cos^2\varphi+\cos^2\beta-2\cos\varphi\cos\beta\cos\beta_1\right], \\[6pt] m_z^2 &= \frac{1}{\cos^2\varphi} \left[\cos^2\varphi+\cos^2\gamma-2\cos\varphi\cos\gamma\cos\gamma_1\right]. \end{aligned} \right\} \tag{1} \]

Denoting by \(\omega_x,\ \omega_y\), and \(\omega_z\) the angles opposite the coordinate axes \(X_1,\ Y_1\), and \(Z_1\) in the projection, and solving the oblique triangles \(AO_1B,\ BO_1C\), and \(CO_1A\), in which the sides \(AB,\ BC\), and \(CA\) are respectively equal to \(\sqrt{x^2+y^2},\ \sqrt{y^2+z^2}\), and \(\sqrt{z^2+x^2}\), after simple transforma-

we obtain the following general formulas determining the angles between the coordinate axes in the projection:

\[ \left. \begin{aligned} \cos \omega_x &= \frac{(m_y^2-1)\cos^2\gamma+(m_z^2-1)\cos^2\beta} {2m_y m_z\cos\beta\cos\gamma}, \\[6pt] \cos \omega_y &= \frac{(m_x^2-1)\cos^2\gamma+(m_z^2-1)\cos^2\alpha} {2m_x m_z\cos\alpha\cos\gamma}, \\[6pt] \cos \omega_z &= \frac{(m_x^2-1)\cos^2\beta+(m_y^2-1)\cos^2\alpha} {2m_x m_y\cos\alpha\cos\beta}. \end{aligned} \right\} \tag{2} \]

The formulas given above for the scales of distortion along the coordinate axes (1) and the general formulas for determining the angles between the coordinate axes in the projection (2) fully resolve the question of constructing any images in parallel projections.

As is not difficult to see from the formulas given, for a given direction of parallel projection one can obtain an infinite set of images, whose form will depend on the relative position of the coordinate axes with respect to the plane of projection; and conversely, for a given position of the coordinate axes and the plane of projection, one can likewise obtain an infinite set of images, whose form will depend on the choice of the direction of projection.

The considerations presented regarding the number of different possible images under parallel projection onto a plane compel one to assume that, from the infinite set of images, it is possible to select those that would satisfy the requirements set forth above: sufficient ease of measurement with the appropriate visual clarity of the image and simplicity of its construction.

In order to make it easier to understand the entire infinitely large number of images obtained under parallel projection onto a plane, when establishing the proper type of image for the given object being projected, the question naturally arises of studying the properties of these images and bringing them into a definite system, i.e., the question of carrying out an appropriate analysis for the purpose of establishing a classification of the images obtained under parallel projection onto a plane.

Classification of projections by method of construction

By the method of construction, as is known, all parallel projections onto a plane may be divided into three classes:

1. The class of trimetric projections.
2. The class of dimetric projections.
3. The class of isometric projections.

The first class includes those projections in which the scales along all three coordinate axes are different, i.e., when

\[ m_x \ne m_y \ne m_z. \]

The second includes projections having equal scales of distortion along two axes, but not equal to the third, i.e., when

\[ m_x = m_y \ne m_z \]

or

\[ m_x=m_z\ne m_y \]

or

\[ m_x\ne m_y=m_z. \]

And, finally, the third class comprises projections having equal distortion scales along all three coordinate axes, i.e., when

\[ m_x=m_y=m_z. \]

Depending on what angle \(\varphi\) the direction of parallel projection forms with the perpendicular to the plane of projection, each class can be divided further into an infinite set of genera; for example, when \(\varphi=0^\circ\) one obtains the so-called genus of orthogonal, or rectangular, projections; when \(\varphi=45^\circ\), the genus of semi-rectangular oblique projections, and so forth.

Besides subdivision into classes and genera, depending on the relative position occupied by the coordinate axes with respect to the plane of projection for a given direction of projection, each genus of projections may further be subdivided into types; for example, taking some genus of projections characterized by the angle \(\varphi\), and setting \(\alpha=90^\circ\) and \(\beta=90^\circ\), we obtain projections of the simplest type, and so forth.

The general formulas derived earlier for the distortion scales along the coordinate axes (1) and the general formulas determining the values of the angles between the axes (2), as formulas that in the general case give different distortion scales along all three axes, are at the same time also the general formulas for the class of trimetric projections.

The class of dimetric projections, besides subdivision into genera and types, may also be divided into three large groups:

1. Group in which \(m_x=m_y\)

2. ” ” \(m_x=m_z\)

3. ” ” \(m_y=m_z\)

The general formulas for the class of dimetric projections, remembering that

\[ \cos^2\alpha+\cos^2\beta+\cos^2\gamma=1 \]

and

\[ \cos\alpha\cos\alpha_1+\cos\beta\cos\beta_1+\cos\gamma\cos\gamma_1=\cos\varphi, \]

we shall determine by the simultaneous solution of any two equations (1), finding the values of the distortion scales through the angles \(\varphi,\beta\) and \(\gamma_1\), \(\varphi,\beta\) and \(\beta_1\), and \(\varphi,\alpha\) and \(\alpha_1\) for all three groups of the class.

And, finally, to obtain the general formulas for the class of isometric projections, by the simultaneous solution of all three equations (1) we express the distortion scales along the coordinate axes through the angle \(\varphi\).

The general formulas characterizing the classes, groups, two principal genera (rectangular at \(\varphi=0^\circ\) and semi-rectangular at \(\varphi=45^\circ\)) and the simplest type of projections, combined into a single summary table, are given below.

CLASSIFICATION
of projections obtained by the parallel method of projection onto a plane

Distortion of the Azimuth of a Direction

To determine the axonometric projection $\alpha'$ of the azimuth $\alpha$ of a certain direction $UT$, let us imagine the axonometric axes transferred to the point $U$ (Fig. 2). Having constructed, from the coordinate increments $\Delta x$, $\Delta y$, and $\Delta z$ (taking into account the distortion scales along the coordinate axes), the projection $t$ of the point $T$ and joining it by the straight line $ta$ with the point $a$, from the triangle $ati$ we find:

\[ \frac{at}{\sin \alpha'}=\frac{m_x\cdot \Delta x}{\sin(\alpha'+j)}, \]

whence, after simple transformations, we obtain the following expression for the azimuth of the direction in projection:

\[ \operatorname{tg}\alpha'=\frac{at\cdot \sin j}{m_x\cdot \Delta x-at\cdot \cos j} \tag{c} \]

Fig. 2

It is not difficult to see (see Fig. 2) that

\[ j=180^\circ-\omega_z-i, \]

where the angle $i$ can be determined after solving the oblique triangle $abt$ in the form

\[ \sin i=\frac{m_z\cdot \Delta z}{at}\sin\omega_x. \]

Using the obtained value of $\sin i$, for $\sin j$ and $\cos j$ we find, after the corresponding transformations, the following expressions:

\[ \sin j=\frac{1}{at}\left[m_y\cdot \Delta y\cdot \sin\omega_z+m_z\cdot \Delta z\sin\alpha_z\right] \]

\[ \cos j=-\frac{1}{at}\left[m_y\cdot \Delta y\cos\omega_z+m_z\cdot \Delta z\cos\alpha_z\right], \]

where $\alpha_z=\omega_z+\omega_x$.

Substituting the values of $\sin j$ and $\cos j$ into formula (c), for $\operatorname{tg}\alpha'$ we obtain:

\[ \operatorname{tg}\alpha'= \frac{m_y\cdot \Delta y\sin\omega_z+m_z\cdot \Delta z\sin\alpha_z} {m_x\cdot \Delta x+m_y\cdot \Delta y\cos\omega_z+m_z\cdot \Delta z\cos\alpha_z} \]

and finally, after dividing the numerator and denominator by $\Delta x$ and remembering that

\[ \frac{\Delta y}{\Delta x}=\operatorname{tg}\alpha \quad\text{and}\quad \frac{\Delta z}{\Delta x}=\frac{\operatorname{tg}\delta}{\cos\alpha}, \]

where $\alpha$ is the strike, and $\delta$ is the dip of the straight line $UT$, we may write the final expression for $\operatorname{tg}\alpha'$ in the form:

\[ \operatorname{tg}\alpha'= \frac{m_y\sin\omega_z\sin\alpha+m_z\sin\alpha_z\operatorname{tg}\delta} {m_x\cos\alpha+m_y\cos\omega_z\sin\alpha+m_z\cos\alpha_z\operatorname{tg}\delta} \tag{3} \]

By the obtained formula (3), for any type of parallel projection, it will not be difficult to compute, from the known strike and dip of a straight line in space, the strike of its image in the projection.

Substituting the value of \(\operatorname{tg}\alpha'\) from equation (3) into the known formula

\[ \operatorname{tg}(\alpha-\alpha')=\frac{\operatorname{tg}\alpha-\operatorname{tg}\alpha'}{1+\operatorname{tg}\alpha\,\operatorname{tg}\alpha'}, \]

we obtain the following expression for the magnitude of the distortion of azimuths:

\[ \operatorname{tg}(\alpha-\alpha')= \frac{ m_x\sin\alpha\cos\alpha+ m_y\sin\alpha\sin(\alpha-\omega_z)+ m_z\operatorname{tg}\delta\,\sin(\alpha-\alpha_z) }{ m_x\cos^2\alpha+ m_y\sin\alpha\cos(\alpha-\omega_z)+ m_z\operatorname{tg}\delta\,\cos(\alpha-\alpha_z) }. \tag{4} \]

The expression (4) found for the magnitude of the distortion of azimuths in a projection is general for all types of parallel projection onto a plane. To obtain the value of the distortion of azimuths corresponding to a given type of projection, it is sufficient to substitute into formula (4) the corresponding values of the distortion scales and the angle \(\omega_z\); thus, for example, for the first group of dimetric projections of the simplest type (with \(m_x=m_y=1\) and \(\omega_z=90^\circ\)) we find:

\[ \operatorname{tg}(\alpha-\alpha')= \frac{m_z\sin(\alpha-\alpha_z)} {\operatorname{ctg}\delta+m_z\cos(\alpha-\alpha_z)}, \tag{5} \]

and for projections of the simplest type of the rectangular isometric class:

\[ \operatorname{tg}(\alpha-\alpha')= \frac{\sin(\alpha-\alpha_z)} {\operatorname{ctg}\delta+\cos(\alpha-\alpha_z)}. \tag{6} \]

Scale of distortion of the length of a straight line of arbitrary direction

Let us call the length of the projected line \(UT\) equal to \(l\), the length of its image equal to \(l_1\), and denote by \(m\) the particular scale of distortion of length. The relation of these three quantities is ordinarily expressed by the formula:

\[ m=\frac{l_1}{l}. \]

From triangle \(atu\) (Fig. 2) we have:

\[ l_1^2=m_x^2\Delta x^2+at^2-2m_x\Delta x\cdot at\cos j \]

or, after substituting the value derived earlier for \(\cos j\):

\[ l_1^2=m_x^2\Delta x^2+at^2+2m_xm_y\Delta x\Delta y\cos\omega_z +2m_xm_z\Delta x\Delta z\cos\alpha_z. \]

Substituting also the value of \(at^2\), which may be obtained from the solution of triangle \(abt\), in the form

\[ at^2=m_y^2\Delta y^2+m_z^2\Delta z^2 +2m_ym_z\Delta y\Delta z\cos(\alpha_z-\omega_z), \]

we obtain

\[ \begin{aligned} l_1^2={}&m_x^2\Delta x^2+m_y^2\Delta y^2+m_z^2\Delta z^2 +2m_xm_y\Delta x\Delta y\cos\omega_z\\ &+2m_xm_z\Delta x\Delta z\cos\alpha_z +2m_ym_z\Delta y\Delta z\cos(\alpha_z-\omega_z). \end{aligned} \]

And finally, dividing both sides by \(l^2\) and remembering that

\[ l^2=\Delta x^2+\Delta y^2+\Delta z^2,\qquad \frac{\Delta y}{\Delta x}=\operatorname{tg}\alpha,\qquad \frac{\Delta z}{\Delta x}=\frac{\operatorname{tg}\delta}{\cos\alpha}, \]

we obtain, for the partial scale of length distortion under the parallel method of projection onto a plane, after certain transformations, the following expression:

\[ \begin{aligned} m^{2}={}&[m_x^{2}\cos^{2}\alpha + m_y^{2}\sin^{2}\alpha]\cos^{2}\delta + m_z^{2}\sin^{2}\delta + m_xm_y\sin 2\alpha \cos^{2}\delta\cos\omega_z \\ &+ m_z[m_y\sin\alpha\cos(\alpha_z-\omega_z)+m_x\cos\alpha\cos\alpha_z]\sin 2\delta \end{aligned} \tag{7} \]

The formula (7) obtained for the partial scale of length distortion, like formulas (3) and (4), is general for all classes of parallel projection. After substituting the corresponding values for \(m_x\), \(m_y\), \(m_z\), and \(\omega_z\), for the first group of dimetric projections we find, in the simplest form:

\[ m^{2}=\cos^{2}\delta+m_z^{2}\sin^{2}\delta+m_z\cos(\alpha-\alpha_z)\sin 2\delta, \tag{8} \]

and for the simplest type of the semi-rectangular genus of the isometric class,

\[ m^{2}=1+\cos(\alpha-\alpha_z)\sin 2\delta. \tag{9} \]

Scale of area distortion

Let us imagine a certain plane whose bedding elements are strike—\(\alpha_p\) and dip—\(\delta_p\). From this plane select a square area \(ABCD\), whose sides shall be equal to unity, and consequently whose area will also be equal to unity. Suppose that the sides \(AD\) and \(BC\) of the square thus taken are directed along the strike of the plane, and the sides \(AB\) and \(DC\)—along its dip. Since, under parallel projection, parallel straight lines are represented by parallel straight lines, consequently the image \(A_1B_1C_1D_1\) of the square \(ABCD\), in the general case, will represent a parallelogram, whose area \(F_1\) can be represented in the form (Fig. 3)

Fig. 3

\[ F_1=\overline{B_1C_1}\times \overline{A_1B_1}\sin v, \]

where

\[ v=180^\circ-(\alpha_2' - \alpha_1'), \]

\(\alpha_2'\)—the strike of the line \(AB\) in the projection,
\(\alpha_1'\)— “ ” \(BC\) “ ”,
\(\overline{A_1B_1}\) and \(\overline{B_1C_1}\)—the lengths of the sides of the parallelogram.

Calling \(m_2\) the scale of length distortion for the line \(AB\), and \(m_1\) the scale of length distortion for the line \(BC\), and remembering that \(AB=BC=1\), we find for the sides of the parallelogram the values:

\[ \overline{A_1B_1}=m_2 \quad\text{and}\quad \overline{B_1C_1}=m_1. \]

Substituting the values found for the sides of the parallelogram and the angle \(\vartheta\) into the formula for the area of a parallelogram, we shall have:

\[ F_1=m_1m_2\sin(\alpha'_2-\alpha'_1), \tag{d} \]

Using the formulas given above for the tangent of the azimuth on the projection (3) and the particular scale of distortion of lengths (7), for the directions \(AB\) and \(BC\), whose elements of occurrence are respectively \(\alpha=\alpha_p+90^\circ,\ \delta=\delta_p\) and \(\alpha=\alpha_p,\ \delta=0^\circ\), we obtain the following set of formulas:

\[ m_1^2=m_x^2\cos^2\alpha_p+m_y^2\sin^2\alpha_p+2m_xm_y\cos\omega_z\sin\alpha_p\cos\alpha_p, \]

\[ \begin{aligned} m_2^2={}&m_x^2\sin^2\alpha_p\cos^2\delta_p +m_y^2\cos^2\alpha_p\cos^2\delta_p +m_z^2\sin^2\delta_p \\ &-2m_xm_y\cos\omega_z\sin\alpha_p\cos\alpha_p\cos^2\delta_p \\ &-2m_xm_z\cos\alpha_z\sin\alpha_p\sin\delta_p\cos\delta_p +2m_ym_z\cos(\alpha_z-\omega_z)\cos\alpha_p\sin\delta_p\cos\delta_p, \end{aligned} \]

\[ \cos\alpha'_2= \frac{-m_x\sin\alpha_p\cos\delta_p +m_y\cos\omega_z\cos\alpha_p\cos\delta_p +m_z\cos\alpha_z\sin\delta_p}{m_2}, \]

\[ \sin\alpha'_2= \frac{m_y\sin\omega_z\cos\alpha_p\cos\delta_p +m_z\sin\alpha_z\sin\delta_p}{m_2}, \]

\[ \cos\alpha'_1= \frac{m_x\cos\alpha_p+m_y\cos\omega_z\sin\alpha_p}{m_1} \]

and

\[ \sin\alpha'_1= \frac{m_y\sin\omega_z\sin\alpha_p}{m_1}, \]

using which we find the value of \(\sin(\alpha'_2-\alpha'_1)\) in the form:

\[ \sin(\alpha'_2-\alpha'_1)= \frac{1}{m_1m_2} \left[ m_xm_y\sin\omega_z\cos\delta_p +m_xm_z\sin\alpha_z\cos\alpha_p\sin\delta_p + m_ym_z\sin(\alpha_z-\omega_z)\sin\alpha_p\sin\delta_p \right]. \]

Substituting the obtained value of \(\sin(\alpha'_2-\alpha'_1)\) into formula (d), for the area of the parallelogram we finally obtain the expression:

\[ F_1=m_xm_y\sin\omega_z\cos\delta_p +m_xm_z\sin\alpha_z\cos\alpha_p\sin\delta_p +m_ym_z\sin(\alpha_z-\omega_z)\sin\alpha_p\sin\delta_p. \]

And finally, calling the scale of area distortion \(m_p\) the ratio of the area of the image of the square \(F_1\) to its actual area \(F\), i.e.,

\[ m_p=\frac{F_1}{F} \]

and remembering that, by condition, \(F=1\), we obtain for the scale of area distortion in general form the expression:

\[ m_p=m_xm_y\sin\omega_z\cos\delta_p +m_xm_z\sin\alpha_p\cos\alpha_p\sin\delta_p +m_ym_z\sin(\alpha_z-\omega_z)\sin\alpha_p\sin\delta_p, \tag{10} \]

Substituting the corresponding values \(m_x,\ m_y,\ m_z\) and \(\omega_z\) for the first group of dimetric projections of the simplest type, we obtain:

\[ m_p=\cos\delta_p+m_z\sin(\alpha_z-\alpha_p)\sin\delta_p, \tag{11} \]

and for the simplest type of the semi-rectangular genus of the isometric class

\[ m_p=\cos\delta_p+\sin(\alpha_z-\alpha_p)\sin\delta_p. \tag{12} \]

Classification of Plane Projections According to the Property of the Image

According to the property of the image, plane projections may be divided into the following four types:

1) equal-area,
2) equiangular,
3) equal-azimuthal,
4) arbitrary.

In constructing equal-area projections, which preserve the true magnitude of the areas of individual figures of the plane being projected, it is necessary, as is known, to satisfy the condition that the area-distortion scale be equal to unity; to this end, equating equation (10) to unity,

\[ m_x m_y \sin \omega_z \cos \delta_p + m_x m_z \sin \alpha_z \cos \alpha_p \sin \delta_p + m_y m_z \sin(\alpha_z-\omega_z)\sin \alpha_p \sin \delta_p = 1, \tag{13} \]

we obtain the relation required for equal-area plane projections between the distortion scales along the coordinate axes \(m_x, m_y, m_z\) and the angles \(\omega_z\) and \(\alpha_z\).

By assigning arbitrary values to four of the unknowns entering condition (13), the solution of the latter gives the fifth unknown, which determines the equal-area character of the projection.

Of the possible equiangular plane projections under parallel projection, we shall consider only the case in which the plane of projection is parallel to the plane being projected, the elements of attitude of which are strike \(\alpha_p\) and dip \(\delta_p\). Obviously, in this case (see Fig. 1) the values of the angles \(\alpha, \beta\), and \(\gamma\), which determine the position of the projection plane in space, will be:

\[ \cos \alpha=\sin \alpha_p \sin \delta_p,\quad \cos \beta=\cos \alpha_p \sin \delta_p \quad \text{and} \quad \gamma=\delta_p, \tag{e} \]

Substituting the values found for \(\alpha, \beta\), and \(\gamma\) into the formulas determining the values of the angles between the coordinate axes in the projection (2), we obtain the following three conditions expressing the relation between the factors that determine the character of the image \((m_x, m_y, m_z, \omega_z\) and \(\alpha_z)\):

\[ \left. \begin{aligned} \cos \omega_x=\cos(\alpha_z-\omega_z) &= \frac{(m_y^2-1)\cos^2 \delta_p+(m_z^2-1)\cos^2 \alpha_p \sin^2 \delta_p} {2m_y m_z \cos \alpha_p \sin \delta_p \cos \delta_p}, \\[6pt] \cos \omega_y=\cos \alpha_z &= \frac{(m_x^2-1)\cos^2 \delta_p+(m_z^2-1)\sin^2 \alpha_p \sin^2 \delta_p} {2m_x m_z \sin \alpha_p \sin \delta_p \cos \delta_p}, \\[6pt] \cos \omega_z &= \frac{(m_x^2-1)\cos^2 \alpha_p+(m_y^2-1)\sin^2 \alpha_p} {2m_x m_y \sin \alpha_p \cos \alpha_p}. \end{aligned} \right\} \tag{14} \]

For projections that preserve the natural magnitude of the azimuths of directions, the relation between the quantities \(m_x, m_y, m_z, \omega_z\), and \(\alpha_z\) will be obtained from the condition that the previously derived value for the distortion of azimuths in the projection be equal to zero.

Equating equation (4) to zero, we find:

\[ m_x \sin \alpha \cos \alpha + m_y \sin \alpha \sin(\alpha-\omega_z) + m_z \operatorname{tg}\delta \sin(\alpha-\alpha_z) = 0 \tag{g} \]

To determine the condition of equal-azimuthality of plane projections, we shall initially set the azimuth of the direction in equation (g) equal to zero and obtain the following particular condition for equal-azimuthality of plane projections:

\[ -m_z \operatorname{tg}\delta \sin \alpha_z=0, \tag{h} \]

whence (since in the general case \(\operatorname{tg}\delta \sin \alpha_z \ne 0\)) we have:

\[ m_z=0. \]

Substituting the found value of \(m_z\) into equation \((g)\), we find:

\[ m_x \sin \alpha \cos \alpha + m_y \sin \alpha \sin(\alpha-\omega_z)=0 \]

or, after division by \(\sin \alpha\):

\[ m_x \cos \alpha + m_y \sin(\alpha-\omega_z)=0 . \tag{i} \]

In order that the last equality \((i)\) remain equal to zero for all values of the azimuth of the direction, as is not difficult to see from the construction of the formula itself, it is necessary that \(m_x=m_y\) and \(\omega_z=90^\circ\).

Thus, for constructing an equi-azimuthal projection the following conditions are necessary:

\[ m_x=m_y,\ \omega_z=90^\circ \text{ and } m_z=0; \tag{15} \]

the angle \(\alpha_z\) is absent, since \(m_z=0\).

In those cases when the factors \(m_x, m_y, m_z, \omega_z\), and \(\alpha_z\) do not satisfy the three conditions set forth above, so-called arbitrary projections will occur.

Choice of Projection in Representing Underground Workings

The existence of an innumerable set of different projections in the parallel method of projection provides great latitude and at the same time makes it difficult to choose the projection most suitable for representing the workings of a given bed.

The most convenient projections for representing underground workings (in the sense of simplicity of construction) will undoubtedly be those in which at least two scales of distortion along the coordinate axes are equal to unity.

Such projections, as is not difficult to see from the formulas given, when projecting onto a horizontal plane of projection, will be projections that do not distort the azimuth and length of horizontal straight lines, i.e. the family of semirectangular projections of the isometric class and all families of the first group of dimetric projections of the simplest type, i.e. such dimetric projections of the first group whose form is determined by the angle \(=0^\circ\), and whose scales of distortion along the coordinate axes are equal to \(m_x=1\), \(m_y=1\), and \(m_z=\operatorname{tg}\varphi\). When projecting onto a vertical plane of projection (i.e. when constructing profiles), it is best to take the family of semirectangular projections of the isometric class or any family of the second group of dimetric projections of the simplest type (for \(\beta=90^\circ\)), for which

\[ m_x^2=\frac{1}{2}(1+\operatorname{tg}^2\varphi),\quad m_y^2=1 \text{ and } m_z^2=\frac{1}{2}(1+\operatorname{tg}^2\varphi). \]

If, within the area of the given planchette, the bed being represented lies calmly, i.e. has no abrupt jumps in the change of its strike and dip, then, taking the average value of its elements of occurrence over the represented part of the bed, it is best to choose an equal-area projection of the simplest type of the rectangular family of the isometric class, as the simplest to construct and the most convenient in terms of calculating reserves of the mineral. The apparatus of formulas necessary for constructing such a projection can easily be obtained from the basic formulas given above, if in the latter one sets \(\varphi=45^\circ\), \(\alpha=90^\circ\), \(\beta=90^\circ\), and \(\gamma=0^\circ\).

Proceeding from formulas (1), for the distortion scales along the coordinate axes we shall have the values:

\[ m_x=1,\quad m_y=1 \text{ and } m_z=1. \]

According to formula (2), for the angle \(\omega_z\) we obtain:

\[ \omega_z=90^\circ. \]

The value of the angle \(\alpha_z\) is obtained from the condition of equal magnitude of the projections given in equation (11), in the form:

\[ \sin(\alpha_z-\alpha_p)=\frac{\operatorname{tg}\delta_p}{2}. \]

The distortion of the azimuths of directions, the value of the length scale, and the area scale for the given projection are determined by the formulas derived earlier, (6), (9), and (12).

For general overviews, when several seams and the ground surface are shown simultaneously, the most suitable projection (from the standpoint of simplicity of construction) will be an equiangular projection of the first group, of the dimetric class, in the simplest form; for it, by a solution analogous to that of the preceding projection, we obtain

\[ m_x=m_y=1,\quad \omega_z=90^\circ, \]

and for the quantities \(m_z\) and \(\alpha_z\), proceeding from the 1st and 2nd equations (14), we find:

\[ \alpha_z=\alpha_p+270^\circ \]

\[ m_z=\operatorname{ctg}\frac{\delta_p}{2}. \]

The value of the distortion of azimuths, the scale of lengths, and of areas is determined by formulas (5), (8), and (11).

The question of constructing the most graphic representation of the workings of a given seam can evidently be resolved in greater detail only practically, since the visual clarity of such representations will depend not only on the elements of occurrence of the seams, but also on the thickness of the seam, as well as on the adopted system of working the mineral deposit.

Methods for constructing the image \(t\) of a spatial point \(T\) in the simplest form of projections of the dimetric and isometric classes

Since, in constructing an axonometric representation of underground workings, the need most often may arise to obtain the representation on a horizontal plane, in what follows we shall take this latter plane as the plane of projection. The axes \(O_1X_1\) and \(O_1Y_1\) on the plane of the paper will then be arranged at a right angle to one another (for the simplest form, \(\omega_z=90\), \(m_x=m_y=1\)), while the axis \(O_1Z_1\) may be arranged arbitrarily.

Let us arrange the axes \(O_1X_1\) and \(O_1Y_1\) as is customary in constructing ordinary plans; that is, let us direct the axis \(O_1X_1\) upward on the sheet of paper (to the north), and the axis \(O_1Y_1\) perpendicular to the axis \(O_1X_1\) to the right (to the east). We draw the axis \(O_1Z_1\) in such a way that at the point \(O_1\) it forms with the axis \(O_1X_1\) some angle \(\alpha_z\), measured clockwise and determined from the conditions of occurrence of the deposit being represented.

When constructing the axonometric image of a point, its position may be specified in three ways:

1. By three coordinates \(x\), \(y\), and \(z\).
2. By the azimuth of the direction \(\alpha\) to the point \(T\) from some second point \(U\), for which the axonometric image \(u\) is already available, by the distance \(l\) between the points \(U\) and \(T\), and by the angle of inclination to the horizon \(\delta\) for the line \(UT\).
3. By the horizontal angle \(\omega\), measured at the point \(U\) between the directions to the points \(T\) and \(V\) (the points \(U\) and \(V\) are assumed already plotted on the projection), by the distance \(l\) between the points \(U\) and \(T\), and by the angle of inclination to the horizon \(\delta\) for the line \(UT\).

To construct the image \(t\) of the point \(T\) under the first method of specification (see Fig. 4), from the point \(O_1\) along the axis \(O_1X_1\) we lay off the segment \(O_1a = x\), and along the axis \(O_1Y_1\) the segment \(O_1b = y\); restoring from the points \(a\) and \(b\) perpendiculars to the axes \(O_1X_1\) and \(O_1Y_1\), at their intersection we find a certain point \(t_0\) (the point \(t_0\) is the orthogonal projection of the point \(T\) onto the horizontal plane), from which we draw a straight line parallel to the axis \(O_1Z_1\), and on the latter lay off from the point \(t_0\) the segment \(t_0t = z_1 = m_z z\). The point \(t\) obtained by such a construction will be the axonometric image of the spatial point \(T\).

Fig. 4

In those cases when the segments \(x\), \(y\), and \(z\) are significant in magnitude and there are already on the sheet of paper axonometric images of certain points whose coordinates are also known, it is necessary to determine, from the given nearest point to the point \(T\), the coordinate increments, imagine at the given point the origin of axonometric coordinates, and carry out the constructions just described, with the sole difference that on the imagined axes one must lay off not the coordinates of the point \(T\), but their increments from the given point.

The construction of the image of a point under the first method of specifying it is simplified still further if, instead of the axonometric axes drawn on the paper, one constructs a coordinate graticule after the example of the coordinate grid used in constructing ordinary plans; that is, draw on the paper, at equal distances from one another, lines parallel to the axonometric axes, with marks indicating the distance from the origin of coordinates.

In those cases when the point \(T\) is specified by the second or third methods, two cases of its position may be conceived:

1. The straight lines \(UT\) and \(UV\) are horizontal (for example, in an adit).
2. The straight lines \(UT\) and \(UV\) make some angle \(\delta\) with the plane of the horizon.

In the first case, under the second method of specification (Fig. 5), it is necessary to draw through the point \(u\) the straight line \(uN\) parallel to the axis \(O_1X_1\), construct at the point \(u\) the straight line \(ut\) at the given azimuth \(\alpha\), and lay off on this straight line the segment \(ut = l\); under the third method of specification (Fig. 6), at the given point \(u\) construct the straight line \(ut\) of length equal to \(l\) and forming with the given straight line \(uv\) the angle \(\omega\).

If, however, the straight line \(UT\) is inclined at an angle \(\delta\) to the horizontal plane, then instead of the angle \(\alpha\) and the length \(l\), which would be projected without distortion if \(UT\) were horizontal, it is necessary to construct a certain angle \(\alpha'\) and length \(l_1 = ml\), determined by formulas (5), (6), (8), and (9), or to draw through the given point \(u\) (Fig. 7) the straight line \(ut_0 = l_0\), forming

with the axis \(O_1X_1\) the given azimuth \(\alpha\); from the point \(t_0\) obtained by this construction, draw a direction parallel to the axis \(O_1Z_1'\), on which, from the point \(t_0\), lay off the distance \(t_0t=\Delta z_1=m_2\Delta z\); the point \(t\) will be the required image of the point \(T\).

Fig. 5

Fig. 6

The quantities \(l_0\) and \(\Delta z\) needed for constructing the point \(t\) are the horizontal projection of the line \(UT\) and the elevation of the point \(T\) above the point \(U\), and are determined by the known formulas:

\[ l_0=l\cos\delta \]

and

\[ \Delta z=l\sin\delta . \]

To plot the image of non-control points, it is best to determine the quantities \(\alpha'\) and \(l\) graphically; for this purpose, using the formulas given, one should first construct the corresponding graphs of the distortion of lengths and azimuths, or use a graph giving both quantities \((\alpha'\) and \(l)\) simultaneously. The construction of such a graph, based on the principles of parallel projection, should be carried out from the following considerations.

Fig. 7

Taking a hemisphere of radius \(R=1\) at an arbitrary scale, cut it by a series of horizontal planes spaced from the horizontal plane passing through the center of the sphere at distances equal to \(R\sin\delta\) (Figs. 8 and 9). As a result of such sections, parallel circles are obtained on the surface of the hemisphere, with radii equal to \(\rho=R\cos\delta\). By joining with straight lines all points of the resulting circles to the center of the sphere, we obtain a series of cones with a common vertex at the center of the sphere, whose generators will be inclined to the horizon at angles equal to the selected angles \(\delta\).

Projecting, by parallel rays that make an angle \(\varphi\) with the perpendicular to the plane of projection, all the cones obtained onto the horizontal plane passing through the center of the sphere, we obtain on the plane

a series of eccentric circles with radii \(\rho = R \cos \delta\), the centers of which will lie on one straight line issuing from the projection of the center of the sphere (the vertices of the cones) and separated from it by distances \(Oc_1\), determined from the right triangles \(OTC\) and \(OCc_1\) (Fig. 8) by the formula:

\[ Oc_1 = R \sin \delta \tan \varphi . \]

Since \(R = l a \tan \varphi\) for projections of the simplest kind is equal to \(m_z\), the expression for \(Oc_1\) may be rewritten in the form:

\[ Oc_1 = m_z \sin \delta . \]

The eccentric circles thus obtained will, evidently, be axonometric images of the bases of the cones, while the image of the vertices of the cones will be the center of the sphere. From the diagram constructed in this manner (Fig. 9) it will be possible at once to determine both the scale of distortion of length and the azimuth of direction on the projection, if one imagines all the circles of the bases of the cones and their images to be graduated through \(360^\circ\) in such a way that the marks corresponding to one and the same number of degrees would lie on the circles of the bases of the cones in one vertical plane (on the projection, the straight lines connecting the centers of the various circles with identical azimuth marks are parallel to one another).

Fig. 8

Fig. 9

In fact, any generator \(OT\) of some cone \(OABT\) may be understood as a straight line inclined to the horizon at an angle \(\delta\) and having a strike azimuth equal to \(\alpha\); on the projection this straight line will be represented by the straight line \(Ot\), where \(O\) is the center of the sphere, and point \(t\) is the point of the image of the circumference of the base of the cone, separated from the origin \(O_1\) by an angular distance equal to \(\alpha\). The length of the straight line \(Ot\) will represent ...

the scale of lengths for directions determined by the azimuth \(\alpha\) and the dip \(\delta\), and the point of intersection \(t_0\) of this straight line with the principal circle \(DE\) will mark on the latter the azimuth \((\alpha')\) on the projection for the direction taken. It is not difficult to verify this, if one recalls the meaning of the partial scale of lengths and of the azimuths of directions.

For the scale of distortion of lengths in the simplest form of a projection of the dimetric class we have (equation 8):

\[ m^2=\cos^2\delta+m_z^2\sin^2\delta+m_z\cos(\alpha-\alpha_z)\sin2\delta; \tag{k} \]

solving the oblique triangle \(Oc_1t\) with respect to the side \(Ot\), we find:

\[ Ot^2=tc_1^2+Oc_1^2+2tc_1\cdot Oc_1\cdot\cos(\alpha-\alpha_z); \]

but, according to what was derived above, \(tc_1=\cos\delta\) and \(Oc_1=m_z\sin\delta\), therefore

\[ Ot^2=\cos^2\delta+m_z^2\sin^2\delta+m_z\cos(\alpha-\alpha_z)\sin2\delta; \tag{l} \]

comparing equalities \((k)\) and \((l)\), we find:

\[ Ot=m. \]

Substituting in equation (3), for \(\operatorname{tg}\)—the \(\operatorname{ctg}\) of the azimuth on the projection, \(m_x=m_y=1\) and \(\omega_z=90^\circ\), for dimetric projections of the simplest form we shall have:

\[ \operatorname{tg}\alpha'= \frac{\sin\alpha+m_z\sin\alpha_z\operatorname{tg}\delta} {\cos\alpha+m_z\cos\alpha_z\operatorname{tg}\delta}. \tag{m} \]

Solving the same triangle with respect to the angle \(c_1Ot\), we obtain:

\[ \frac{tc_1}{\sin\angle c_1Ot} = \frac{Oc_1}{\sin[\angle Oc_1t+\angle c_1Ot]}. \]

Or, since \(tc_1=\cos\delta\), \(Oc_1=m_z\sin\delta\), \(\angle Oc_1t=180^\circ-(\alpha_z-\alpha)\), and \(\angle c_1Ot=\alpha_z-\angle X_1Ot_0\):

\[ \frac{\cos\delta}{\sin(\alpha_z-\angle X_1Ot_0)} = \frac{m_z\sin\delta}{\sin(\angle X_1Ot_0-\alpha)}, \]

whence

\[ \operatorname{tg}\angle X_1Ot_0= \frac{\sin\alpha+m_z\sin\alpha_z\operatorname{tg}\delta} {\cos\alpha+m_z\cos\alpha_z\operatorname{tg}\delta}; \tag{n} \]

comparing equalities \((m)\) and \((n)\), we find:

\[ \angle X_1Ot_0=\alpha'. \]

Thus, to determine the scale of distortion of length \(m\) and the azimuth of the direction on the projection \(\alpha'\), we find on the graph the circle corresponding to the angle of inclination \(\delta\); on the circle found we locate the point \(t\), separated from the origin \(O_1^\circ\) by an angular distance equal to \(\alpha\); we join point \(t\) with point \(O\) by the straight line \(tO=m\); we measure the length of the straight line \(tO=m\); we determine the point of intersection \(t_0\) of this straight line with the principal circle and, finally, determine the position of point \(t_0\) on the principal circle, i.e., we find the azimuth of the direction on the projection \(\alpha'\).

Conclusion

Comparing the method of orthogonal projection onto a horizontal projection plane adopted for constructing plans (according to the classification given, this projection belongs to the first group of the dimetric class, of the rectangular kind, of the simplest type) with the method of oblique parallel projection of the dimetric and isometric classes, one may note that the advantage of the first method is the invariance of the azimuths of directions. At the same time, this method, which makes visible only two dimensions, has a major drawback: first, the low visual clarity of the representation, and second, the impossibility of depicting on the plane three-dimensional forms situated one above another in horizontal planes. In contrast, the second method described above, while having such shortcomings as distortion of the azimuths of directions and considerable variation in partial scales, has the advantage of preserving good visual clarity of the represented forms.

If one further takes into account that the use of numerical elevations in the simplest type of projections of the dimetric and isometric classes (proposed by me in the work “The Foundations of the Method of Numerical Elevations in the Simplest Type of Parallel Projections of the Dimetric and Isometric Classes”) makes it possible to solve, with comparative ease, all problems connected with determining the natural magnitude of distorted forms and problems of design, then one may hope that this method will be assigned its proper place in solving various kinds of problems and, above all, in the mining industry, in the representation and design of underground workings of a mineral deposit in steeply dipping and thick seams.