On the Construction and Use of Plans in Projection onto an Inclined Plane

A. I. Volkov, P. A. Marchenko

(Presented by the scientific seminar of the departments of mine surveying and geodesy)

The merit of any mine-surveying plan is determined by the degree of clarity of the representation and by the convenience of its use for solving various technical problems.

These qualities of a plan depend on the character of the occurrence of the deposit being represented and on the selected plane of projection. Indeed, when a steeply dipping deposit is represented on a horizontal projection plane, a considerable distortion of the form of the deposit occurs; for this reason such a plan is not visual and is inconvenient for practical use. An analogous situation is obtained when gently dipping deposits are represented on a vertical projection plane.

In order to eliminate the indicated shortcomings, in mine-surveying literature it is recommended in individual cases to compile plans in projection onto an inclined plane. A deposit having regular elements of occurrence is projected onto an inclined plane parallel to the deposit without distortion; consequently, the linear quantities on such a plan will correspond to their true values, and therefore the use of such plans for production purposes is very convenient. The advantage of such a projection diminishes when more complex forms of deposit are represented, and with very complex forms such an advantage disappears altogether.

For constructing projections onto an inclined plane, several methods are recommended in the literature.

Some authors recommend constructing such projections by coordinates obtained as a result of analytical or graphical recalculation of the ordinary coordinates of points. This method is very labor-intensive, since it requires a large amount of computational work or graphical determinations. For this reason, the method of constructing projections onto an inclined plane by coordinates has not found application in practice.

The graphical and grapho-analytical methods of construction currently recommended are not simple and, moreover, have certain other substantial shortcomings.

In the article by V. I. Kuzmin [1], a graphical method is proposed for rearranging plans from a horizontal projection plane onto an inclined projection plane and onto...

inclined one. To construct the projection of a single point onto an inclined plane, this method requires the analytical or graphical determination of five additional quantities \((Z_a - Z_p), Z'_a, f, g, d\). The determination of so many additional quantities complicates the construction and increases the probability of gross errors. In addition, this method also contains a number of other serious shortcomings.

1. In [1] the author states that the quantity \(d\) is always laid off in the direction away from the “horizontal of rotation” of the plane \(P\).

This assertion is valid only for the ideal case in which the points being projected lie on the plane \(P\) itself. In practice, the plane \(P\) is chosen at some distance from the deposit, and on condition that it not intersect the latter anywhere over the entire area of the plan [1]. In this case, practically always, points lying above the plane of projection, when projected onto the inclined plane, will be displaced in the direction of the rise of the plane \(P\), while points lying below the plane \(P\) will be displaced in the direction of its fall. Consequently, the quantity \(d\) must be laid off, in the first case, in the direction of rise, and in the second case in the direction of fall of the plane \(P\) from the projections of these points on the horizontal plane (Fig. 1, a).

Fig. 1.

1. The author gives no recommendations for determining the positions of points in the inclined plane whose horizontal projections lie on the “horizontal of rotation” of the plane \(P\). Analysis shows that the displacement of such points is also subject to the regularity indicated above (Fig. 1, b).

Fig. 2.

1. The formulas proposed by the author for determining the quantity \(d\) are not valid for all cases. For the case when the projection of point \(A\) onto the horizontal plane (point \(a\)) lies from the “horizontal of rotation” in the direction of fall of the plane \(P\) (Fig. 2), the formulas proposed by the author are not applicable.

In order that the formula

\[ d = (Z_a - Z_p)\sin\delta + l(\sec\delta - 1) \tag{1} \]

be valid for all cases of the position of the point being projected, the quantity \(l\) must be regarded as positive if it is measured from the “horizontal of rotation” in the direction of rise of the plane \(P\), and negative—in the direction of fall, taking into account the signs of \(Z_a\) and \(Z_p\). For

formula (1) can be simplified, bearing in mind that \(Z_p = l \cdot \operatorname{tg}\delta\), and represented in the following form

\[ d = Z_a \sin \delta - l(1 - \cos \delta). \tag{2} \]

The advantage of this formula is that it reduces the number of additional quantities to be determined. In accordance with this transformation, the formula for determining the value \(Z_a''\) will have the form:

\[ Z_a'' = Z_a \cos \delta - l \cdot \sin \delta. \tag{3} \]

To eliminate the shortcomings indicated in items 1, 2, and 3, one may, moreover, recommend taking as the “axis of rotation” of the plane \(P\) not its middle horizontal, as the author proposes in [1], but a horizontal lying outside the limits of the deposit being projected.

Docent M. V. Korzhik in [2] proposes a graphic-analytical method for solving this problem. This method also is not distinguished by simplicity, since it requires the graphical or analytical determination of as many as five additional quantities

\[ (Z_a - Z_p),\quad Z_a'',\quad l',\quad \delta y,\quad \Delta Y''. \]

Docent M. V. Korzhik’s proposal to construct projections of points onto an inclined plane from the nearest coincident horizontal of the plane \(P\) eliminates the shortcomings inherent in V. I. Kuzmin’s method; however, the presence of an additional coordinate grid leads to excessive overloading both of the plan and of the projections onto the inclined plane, which is undesirable.

Graphical method for constructing the projection of a point onto an inclined plane from its projection onto a horizontal plane

Plans in projection onto an inclined plane are auxiliary graphical materials and do not preclude the construction of plans in projection onto a horizontal plane, which are obligatory for every mining enterprise. Thus, the desire to use them for constructing projections onto an inclined plane, in order to avoid recalculating coordinates, is understandable.

The method we propose is based on the use of such plans, compiled in projection with numerical elevations. The essence of the proposed method consists in constructing vertical sections passing through the points to be projected onto the inclined plane, across the strike of the selected projection plane \(P\). On these sections the trace of the plane \(P\) and the projected points are shown. In practice, as will be seen below, it is quite sufficient to construct a single summary section-nomogram in order to carry out the transition from projections onto the horizontal plane to projections onto the inclined plane, without resorting to other graphical or analytical determinations.

The recomposition of plans by this method may be carried out in the following order. On the plan compiled in projection onto the horizontal plane, horizontals of the selected plane \(P\) are drawn (Fig. 3). On a separate sheet of stiff paper a vertical section-nomogram is constructed across the strike of the plane \(P\), on which are shown the trace of the plane \(P\), the trace of the vertical plane \(V\), passing-

passing through the “axis of rotation” of the plane \(P\), and lines of the height horizons (Fig. 4). This section should be constructed at the scale of the plan. It is best to construct the projection on tracing paper. On the tracing paper it is sufficient to draw the “axis of rotation” of the plane \(P\) and, for tying it to the plan, either the coordinate grid of the plan or an arbitrarily chosen point on the “axis of rotation” marked on the plan.

Fig. 3.

Fig. 4.

The tracing paper prepared in this way is placed over the plan, and the position of the projected points in the new projection is determined as follows:

a) with a compass, a segment \(l\) is taken from the plan, equal to the perpendicular dropped from the point in the plan to the “axis of rotation”;

b) the segment \(l\) is laid off on the section-nomogram (Fig. 4) from the line \(V\) on the horizon equal to the elevation of the point being projected, on the corresponding side of the “axis of rotation”; as a result we obtain the point \(B\), which is projected orthogonally onto the line \(P\) (the trace of the projection plane) at the point \(b_1\);

c) the distance \(o—b_1\), taken with a compass from the section-nomogram, is laid off on the tracing paper from the “axis of rotation” along the perpendicular passing through the projected point of the plan \((b)\), in the direction of the rise or dip of the plane \(P\), depending on the position of \(b_1\) on the nomogram. As a result we obtain the point \(b_2\), which is the projection of the point \(B\) onto the inclined plane. The segment \(B—b_1\), measured on the section-nomogram, is the “elevation” of the point \(B\) relative to the plane \(P\). The magnitude of this segment in meters is written as the numerical elevation at the point \(b_2\).

In constructing mine workings, the points being projected are characteristic points of these workings. In recasting hypsometric plans, the characteristic points are chosen on the contour lines. In this way any plan can be recast (one representing both the form and the qualitative features of a deposit), provided that elevations have first been determined for the characteristic points.

From the description of the method proposed by us it follows that it is considerably simpler than those recommended earlier, since it does not require any additional analytical or graphical determinations, and evidently this eliminates gross construction errors possible in the previously proposed methods; therefore it can be widely used in mine-surveying practice.

Transition from projection onto an inclined plane to projection onto a horizontal plane

The need to carry out a transition from projection onto an inclined plane to projection onto a horizontal plane arises very rarely in practice. However, when solving certain production problems on the basis of projection onto an inclined plane, it is often necessary

it is necessary to know the position of individual points in projection onto the horizontal plane. Such problems include the construction of a vertical section along a diagonal direction, determination of the azimuth of a straight line, etc.

Fig. 5a

Fig. 5b

The transition indicated above can be carried out with the aid of a section nomogram. For this purpose, from the ends of the line \(a_2-b_2\) (Fig. 5a), given in projection onto the inclined plane, perpendiculars are dropped to the “axis of rotation.” By laying off the lengths of these perpendiculars from the point \(O\) along the line \(P\) (Fig. 5b) in the corresponding direction, the points \(a_1, b_1\) are obtained. From the points \(a_1, b_1\), perpendiculars are dropped to the line \(P\), on which the numerical “elevations” of the points \(a_2, b_2\) are laid off. As a result, the points \(A\) and \(B\) are obtained, whose absolute elevations are determined from the lines of the elevation contours. The distances measured from the points \(A\) and \(B\) to the line \(V\) horizontally are laid off from the “axis of rotation” along the perpendiculars passing through the points \(a_2\) and \(b_2\). The points \(a\) and \(b\) obtained in this way are the projections of the points \(A\) and \(B\) onto the horizontal plane.

Use of Projections onto an Inclined Plane for Solving Certain Mining-Geometrical Problems

In recent times, rather great attention has been given in the mine-surveying literature to the question of constructing projections onto an inclined plane; the question of using these projections for solving various production problems, however, has in fact not been covered in the literature at all. Although this projection makes it possible to solve all problems that arise in practice, analysis shows that their solution requires a special method, different from the method of solving these same problems on plans compiled in projection onto the horizontal plane.

This article considers the solution of problems most frequently encountered in mine-surveying practice.

Construction of a Vertical Section Across the Strike of the Projection Plane

For deposits with uniform elements of occurrence, a section across the strike of the projection plane will at the same time be a section across the strike of the deposit. The need to construct such sections arises rather often in practice. The method of constructing them may be as follows.

Along the section line \(1—1\), marked in projection onto the inclined plane (Fig. 6), the profile of the plane \(P\) and the trace of the horizontal plane \(H\), passing through the “axis of rotation” of the plane \(P\) (Fig. 7), are constructed. The line \(H\) will serve as the reference horizon for

determining the absolute elevations of points. Along line I—I, from the “horizon of rotation” to the characteristic points \(a_2, b_2, c_2\), etc., distances are measured and laid off on the corresponding side from point \(O\) on the section along line \(P\) (Fig. 7). From the ends of these segments (points \(a_1, b_1, c_1\), etc.) perpendiculars are erected to line \(P\), on which the “elevations” of points \(a_2, b_2, c_2\), etc., are laid off upward from line \(P\) if the “elevations” are positive, and downward if they are negative.

As a result of such a construction, points \(A, B, C\), etc., are determined; by joining them with a smooth curve, one obtains the contour of the lying or hanging wall of the deposit in section. The absolute elevations of the points of this contour are determined in the usual way from the reference horizon.

Construction of a vertical section in a diagonal direction

The method described above for constructing a vertical section cannot be used in the present case, since it permits construction only of sections normal to the plane of projection. A vertical section across the strike of the plane is at the same time a normal section to this plane, which is not characteristic of a vertical section in a diagonal direction. The latter is convincingly shown in Fig. 6. In this figure, points 1, 2, 3, 4, etc., lie on the line of section in plan. The projections of these points onto the inclined plane \(P\) are points \(1''\), \(2''\), \(3''\), \(4''\), \(5''\), which, as is evident from the drawing, do not lie on the line of section, but are situated to the side of it along a certain curve, whose character depends on the distance of the section points from plane \(P\). It follows from this that, in order to construct a vertical section in a diagonal direction, it is necessary, for the points lying on the line of section, to determine the values of their “elevations” by means of additional constructions.

The construction of a vertical section in a diagonal direction may be carried out as follows. On the plan in projection onto the inclined plane, the section line II′—II′ is marked (Fig. 6), and by the method described above its projection II—II onto the horizontal plane is determined from any two points lying on line II′—II′. On line II—II, points \(a, b, c\), etc., are marked, on the basis of which the section will be constructed. To determine the elevations of these points in the vertical section, auxiliary vertical sections across the strike of plane \(P\), passing through points \(a, b, c\), are constructed. On the sections obtained in this way (section along 1—1, 3—3, Fig. 8), from point \(O\) along line \(H\) in the corresponding direction,

segments \(a—a'\), \(b—b'\), etc., taken from the plan, are laid off (Fig. 6). From the points \(a, b, c\) on the sections along I—I, II—II, etc., perpendiculars are erected to the line \(H\) and prolonged until they intersect the constructed profile of the deposit at the points \(A, B, C\). For these points the absolute elevations \(Z_a, Z_b, Z_c\), etc., are determined from the reference horizon \(H\); these are the required elevations of the points \(a, b, c\), marked on the section line II—II (Fig. 6). Using the elevations obtained, the vertical section along line II—II is constructed in the usual way.

Fig. 6.

A vertical section along the strike of the deposit (the plane \(P\)) is a special case of a section in a diagonal direction and is constructed analogously to the latter.

Construction of Sections Normal to the Deposit (to the Plane \(P\))

A section normal to the plane \(P\) is, in the general case, not vertical. The exception is a section across the strike of the plane \(P\), since in this case the normal plane coincides with a vertical one. The method for constructing normal sections in any prescribed direction is the same as the method for constructing a vertical section across the strike of the plane \(P\).

Determination of the True Length, Angle of Inclination, and Direction Angle of a Straight Line

If the points \(A\) and \(B\) lie in the projection plane \(P\), or in a plane parallel to it, then the true length between these points is determined by directly measuring the distance between their projections on the inclined plane \(P\). In all other cases, the true length of the straight line between the given points is determined by the usual method, when the elevations of these points are known.

The direction angles of straight lines, when the latter are projected onto an inclined plane, are distorted. The exception to this is straight lines whose direction coincides with the line of dip of the plane \(P\). This distortion occurs as a result of the difference in the “elevations” of the ends of these straight lines relative to the plane \(P\) and of their different distances from the “horizon of rotation”; consequently the projected points of the straight lines are displaced from the “horizon of rotation” of the plane \(P\) by different distances.

To determine the true direction angle of the straight line \(A—B\) (given by the points \(a_2, b_2\)), it is necessary to construct the position of this straight line in projection onto the horizontal plane (Fig. 5 \(a, a—b\)). The angle \(\alpha\) between the \(X\)-axis and the straight line \(a—b\) will be the true direction angle of the straight line \(A—B\).

The angle of inclination of a straight line given in projection on the inclined plane by the points \(a_2, b_2\) is determined from the vertical section constructed along the line \(a—b\).

Determining the True Value of an Area

In most cases, when a deposit is represented in projection onto an inclined plane, the distortion of the surface depicted is insignificant, and the area can then be determined directly from the drawing without taking its distortion into account.

In those cases where the distortion reaches a considerable magnitude, i.e., when the angles of dip and strike of the deposit are not maintained, the distortion is taken into account in the usual way, provided that the “angle of dip” of the deposit in this case is taken to be the angle that it forms with the projection plane \(P\). This angle can be determined by constructing a normal section.

From consideration of the graphical method we propose for constructing projections onto an inclined plane, it is evident that this method is simple and can thus facilitate the introduction of this projection into practice.

The proposed section-nomogram makes it possible not only to carry out very simply the transition from a plan to a projection onto an inclined plane, but also makes it possible, likewise simply, to solve the inverse problem.

It should be noted that the question of solving the inverse problem has not been covered in the literature up to now. The consideration of this question in the present work has enabled us to propose a method for solving a whole series of problems that inevitably arise when working with this projection, which should also contribute to the successful application of this projection in practice.

References

1. Mine Surveying. Collection of scientific works of the Department of Mine Surveying and Geodesy of the Donetsk Industrial Institute, issue IV, Metallurgizdat, 1956.
2. Scientific Reports of Higher Education Institutions (Mining), Sovetskaya Nauka, No. 1, 1958.