ON THEORETICAL STUDIES IN THE FIELD OF AXONOMETRY

V. A. Voskresensky

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

The Great October Socialist Revolution provided every opportunity for the development of science, which became the property of the people. The Soviet government, restoring industry destroyed by the World and Civil Wars, outlined a plan for the further rise of the country’s industrialization and, in a short time, achieved great successes in organizing the training of engineering and technical personnel needed by the various branches of the national economy. To accomplish the assigned task, the number of secondary and higher technical educational institutions was increased; new departments were created both in specialized and in general technical disciplines, in particular in descriptive and engineering graphics. In these disciplines, postgraduate study was opened at the major technical institutes of the Union, making it possible to conduct scientific research fully corresponding to the level of industrial production.

New books and articles appeared in which, unlike the older works—mostly textbooks on methods of practical construction of images—great attention was paid to theory, the importance of which increased still more in connection with the emergence of new branches of science and technology. In the field of developing the theory of axonometric projections, the so-called “fundamental proposition” of axonometry, “both in parallel and in central projection,” was subjected to thorough investigation [1], (p. 4).

Much credit for this belongs to Professor A. K. Vlasov of Moscow State University, who drew the attention of geometers to the development of new theoretical problems of descriptive geometry in general and of axonometric projections in particular.

In 1917 he proposed a new proof of Pohlke’s theorem, which appeared in print only in 1925 [2]. The distinctive feature of this proof was that it was based on the use of affine correspondence, the theory of which served as a foundation for solving various problems of descriptive geometry, and “the fundamental theorem of axonometry—Pohlke’s theorem—received its simplest proof precisely by means of the properties of affine figures”¹).

It should be noted that the proof proposed by Professor A. K. Vlasov becomes impossible in the special case of projecti—

¹) See [3], (p. 3).

when one of the faces of the projected tetrahedron degenerates into a straight line.

The subsequent use of the properties of affine correspondences for the further improvement of the theory of axonometry was carried out by N. A. Glagolev, who on December 9, 1924, delivered a report at the Moscow Mathematical Society on the topic “A Generalization of Pohlke’s Theorem” [4].

Extensive investigations in the theory of axonometric projection were carried out by Professor N. F. Chetverukhin, who in 1933 proposed a theorem on the projection of two centered rectangular systems of axes both under parallel and under central methods of projection [5]. For such a case Pohlke’s theorem gives no answer, since it considers only one system of axes. But since the image of the latter determines the projection, the second system associated with it cannot be represented arbitrarily. Before proceeding to prove the new theorem, N. F. Chetverukhin investigated the possible regions of projection of rectangular coordinate axes situated in space and projected along a preselected direction \(O'O\) (where \(O'\) is the center of the system of axes, and \(O\) is the center of the triad of lines). On the basis of the property of orthogonal projection, the author proved the theorem that “a triad of lines can be regarded as a parallel (central) projection, in the direction \(O'O\), of a rectangular system of axes in space if and only if each divided dihedral angle of the triad of this direction \(O'O\) is obtuse (and each undivided angle is acute).” As a result of the investigations carried out, he established that the boundary surface of the “direction of projection is a cone of the second order” and that “only those lines \(O'O\) which pass outside the boundary cone can serve as the direction of projection.”

Thanks to the determination of the region of projectability of triads of lines, the problem was solved of finding the conditions under which two centered triads of lines lying in some plane are a parallel projection of two rectangular systems of axes in space. The necessity of the condition in this case consists in the fact that the two centered triads of lines contain no separated pairs. The sufficiency of the condition consists in the presence of at least two separated pairs of the triads under consideration. Thus, N. F. Chetverukhin’s investigation of triads of lines is of interest in the following respects:

1. It establishes definite regions of projection of an axonometric system (which Pohlke’s theorem does not provide) and makes it possible to determine in which cases a given triad of lines passing through one point is a parallel (central) projection of a rectangular system of natural coordinate axes in a prescribed direction.

2. The proposed theorem makes it possible to solve the problem of two centered triads of lines as a parallel projection of two rectangular systems of axes, which is very important when one has to deal with the transformation of coordinates, with a moving and a fixed system in mechanics, etc.

3. The theorem on triads of lines makes it possible to examine in greater detail the method of a single image on a plane, chiefly with respect to its visual clarity.

In the course of subsequent investigations N. F. Chetverukhin turned his attention to the geometric correctness of images, which affects the visual clarity of an object. In one of his first works [6], published in 1934, he noted that, in constructing images, one should take into account which elements of it may be chosen arbitrar—

but with such a calculation that this choice fully determines the projection. The proposal stated by the author was developed and improved over the course of several years, and as a result he created an entirely new theory of completeness of images. Proceeding from the geometric correctness of visual images, the author came to the conclusion that any images may be regarded as complete, incomplete, and even overcomplete. Complete images are those on which all incidences of the original can be determined. If it is impossible to establish these, the image is considered incomplete. One of the works in which the completeness and incompleteness of images is touched upon is the article by N. F. Chetverukhin [7], published in 1944; but since the chief question there was the proof of the “affine rigidity of polyhedra,” no complete investigation of the images mentioned and of their metric determinacy was carried out there. All these questions were set forth in great depth in his works [8, 9, 10], published in 1946 and 1947.

In these writings it is considered how a given image determines the original and reflects the positional and metric properties of the latter. Every image (drawing) determines, to varying degrees, the common elements of the lines and surfaces (incidences) of the original. Depending on how many incidences of the original are represented in the image, it may be classed as a complete or incomplete image. Thus, a drawing is called complete if all incidences of the original are determined on it; moreover, those which are not shown in the image can be constructed by using those already present. Consequently, they cannot be chosen arbitrarily, since otherwise overcomplete images may be obtained. The latter may be correct or incorrect, depending on whether or not an arbitrarily chosen superfluous incidence coincides with the incidence obtained by construction.

Let us consider the following example [11].

In a regular quadrilateral prism, through the midpoints of two consecutive sides of the base, a plane has been drawn which intersects three lateral edges and is inclined to the base (Fig. 1). Examining this drawing, we find that it is incorrect.

It is easy to convince oneself of this if one extends, for example, the side \(DC\) to its intersection with the trace \(AB\) of the cutting plane \(ABCDE\).

The line \(AB\) lies in the base plane \(KLM\), which intersects the second base plane \(L_1LM\) along the line \(LM\). Consequently, the point of intersection of the line \(AB\) with the plane \(L_1LM\) must lie on the continuation of the line of intersection of the base planes. At the same time, the line \(CD\), lying in a vertical plane, must meet the plane \(KLM\); and since both lines \(AB\) and \(CD\) belong to the plane \(ABCDE\), they must necessarily intersect at a common point \(P\) on the continuation of the line \(LM\). But, as is evident from the construction, no such point is obtained for the given drawing.

Fig. 1

The error here is that the draftsman arbitrarily assigned superfluous points of the section, thereby making the image overcomplete and incorrect. To construct the correct section, it was necessary, besides the points \(A\) and \(B\) specified by the condition, to take arbitrarily the highest point

on the edge \(MM_1\)—the point \(D\), then find the lines of intersection of the plane \(AB\) with the base planes by joining the points \(P\) and \(Q\) with \(D\), and the figure of the section of the prism would have been constructed correctly (sec. \(ABRDS\)).

An image is called incomplete if it has an insufficient number of incidences for constructing all the remaining incidences of the original. To determine the missing incidences, N. F. Chetverukhin introduced the so-called coefficient of incompleteness \((k)\), by which is meant the number of parameters, or numerical values, having one or another geometrical meaning. In order to find the coefficient of incompleteness for a certain image \(\Phi\), the author uses the “method of successive extension” of a complete image, which consists in the fact that to a system of 4 points representing a tetrahedron (a complete image), one successively adjoins \(1, 2, 3, \ldots, k\) points of the given image. The system of \(n = k + 4\) points is called the point basis of the latter. In [10] the author investigates the parametric computation of complete images in connection with their metric determinacy. By metrically determinate images are meant such images whose original is determined up to similarity. They admit no arbitrary elements, “since there correspond to them completely determinate constructions in the original” (p. 192). But until the complete image is metrically determined, it admits arbitrariness in the metric operations performed on it. This depends on the “supply of free parameters that must be specified in order for the image to become metrically determinate” (p. 192). The number of such a “supply of parameters” is called the “parametrization of the image.” It must be said that there is an essential difference between a complete image and a metrically determinate one. A complete image, according to the definition, makes it possible to find by construction any incidences of its elements. But the original of a complete image remains just as indeterminate as the center of projection \(S'\) and the position of the principal planes \(\sigma'_1\) and \(\sigma'_2\).

To establish the position of the center \(S'\), it is necessary to specify three parameters (the coordinates of the center). To determine the position of the principal plane \(\sigma'_1\), two parameters are required (the angles of inclination of the traces of the plane \(\sigma'_1\) on two coordinate planes to the corresponding axis). The position of the plane \(\sigma'_2\) is given by one parameter (the angle of its inclination to the plane \(\sigma'_1\)). As a result, the parametrization of the complete image is \(p = 6\) (\(p\) is the parametric number). For a metrically determinate image it is sufficient that the difference between \(p\) and the number of specified parameters (introduced with the aid of conditions imposed on the original) should become zero.

In parallel projection, its center is at the improper point \(S'_\infty\), for whose determination two parameters are required. Then the parametric number of complete images in parallel projection is \(p = 5\). If, however, the center of projection is specified, which corresponds to the case of parallel projection with a given direction of projection, the minimal value of the parametric number is \(p = 3\). In parallel projection, the basis for solving metric determinacy is provided by the Pohlke–Schwarz theorem, which N. F. Chetverukhin calls the first existence theorem. Such a name is quite understandable, since it follows from the theorem that “under an arbitrary metric determination of some tetrahedron (up to similarity) there always exists a projection (the direction of projection and the position of the tetrahedron relative to the image plane) that gives an image of it coinciding with the one prescribed in advance” (p. 216). In this case the original, the points of wh—

... whose image is affinely determined with respect to the aforementioned tetrahedron is called metrically determined.

On the basis of the Pohlke–Schwarz theorem, Professor N. F. Chetverukhin proposed a “second existence theorem,” in which a complete quadrilateral with a segment joining one of its vertices to an arbitrary point lying inside the triangle formed by the other three vertices was considered as a parallel projection of a tetrahedron having a right trihedral angle and an altitude dropped from its vertex to the opposite face.

The second existence theorem found application in the following cases:

a) in constructing the perpendicular dropped from the vertex of a trihedral angle to the opposite face, and

b) in constructing plane sections passing through a given point perpendicular to a given line.

Both of these points can be reduced to one general conclusion: the theorem mentioned is applicable whenever it is necessary to carry out the construction of a spatial conjugate pair¹.

Along with theoretical investigations into complete and incomplete images, the author showed the possibility of practical use of the latter, since they possess a large reserve of free parameters. He considered the following cases of application of incomplete images:

a) the image of spatial conjugate pairs;

b) the image of projection onto an axis;

c) the image of the operation of projecting a figure onto a plane.

The theory of complete and incomplete images and the parametric method of constructing them introduced the necessary clarity into questions of the visual character and correctness of the execution of drawings under the conditions of the pedagogical process. This theory is at present finding application in stereometry, analytic and descriptive geometry. Perfecting it, N. F. Chetverukhin in 1947 set forth more fully all the theoretical investigations he had previously expressed in the form of four separate articles [9, 10, 12, 13], supplementing them with special practical applications to projection drawings. His investigations in the field of the theory of visual images are of interest in that they make it possible to construct figures correctly on a plane; they establish the positional properties of the original, as well as its metric determinacy, and find corresponding practical use for all types of projection drawings, including axonometric ones.

Simultaneously with N. F. Chetverukhin, the Leningrad geometer O. A. Vol'berg worked on questions of the theory of axonometry. His investigations, published in 1947 [14], also solve the question of the correctness of constructing images having incidences corresponding to the original. O. A. Vol'berg introduced the concept of monoprojection, by which one should understand an image obtained by projecting a figure from some center onto a given plane, together with additional information about the original attached to the drawing in the form of a description (although the definition does not establish what the nature of this information is). Using affine correspondence, the author set forth the Pohlke–Schwarz theorem and considered a number of metric problems in axonometry.

All theoretical investigations in the field of axonometric projections are reduced mainly to a definite aim—the correctness of con—

constructing one image or another in accordance with a given original and projecting apparatus (i.e., the plane of projection, the center, or the direction of projection). This is the so-called first and principal problem of axonometry. From it, however, there follows directly a second problem—the determination of the original from a given projection and projecting apparatus. There is a complete interconnection between these problems. The Pohlke–Schwarz theorem fully confirms this, since it makes it possible to obtain a reconstruction of the tetrahedron of the original from its parallel projection and, conversely, from a tetrahedron-original specified in advance, to obtain the corresponding projection of a nondegenerate complete quadrilateral. The solution of the inverse problem, encountered in all kinds of manuals and theoretical studies, is rather complicated. Naturally, the question arose of finding a simpler solution for reconstructing a natural coordinate system in space from its given axonometric image. Such a solution was proposed by Scheffers in his textbook of descriptive geometry, published in 1919 [15].

In our literature this question is set forth most fully by Professor E. A. Glazunov, on the basis of the materials of his report delivered at a meeting of the Moscow City Seminar on Descriptive Geometry on March 18, 1949 [16, pp. 35–38].

At present in the USSR there is very extensive material on the theory and practice of axonometric projection, several times surpassing the pre-revolutionary educational and practical manuals that expounded this subject not only in volume but also in depth of investigation.

The authors of numerous articles and dissertation works have engaged in studies of:

a) the theory of constructing axonometric images under parallel projection;

b) methods of solving problems in axonometry;

c) the practical application of axonometric projections in engineering and spatial statics;

d) the mechanization of drafting axonometric images.

The appearance of a large number of works on axonometry in the Soviet period falls in the 1930s and especially the 1940s and 1950s. Such attention to questions of constructing visual images under parallel projection is explained by the creation of new branches of industry, such as automobile, aviation, and others, which require the manufacture of complex structures, the quality of whose production also depends on a correctly composed working complex drawing. Often, owing to the complexity of such drawings, for a faster and clearer understanding of general views, diagrams, and the like, it is necessary to use visual images, predominantly in axonometric projections.

Axonometric projections have found wide use in mining, geological exploration, and civil engineering for depicting block diagrams, profile sections of deposits, mine workings, floor structures, complex building joints, and so on (Figs. 2, 3, 4, 5)¹⁾.

Alongside parallel projection, as is known, there is also central projection, which, in terms of the time of its origin, much earlier found application in painting and architecture in the form of the science of perspective. However, the question of using axonometric

¹⁾ See P. A. Ryzhov, Geometry of the Subsurface. Ugletekhizdat, 1952.

Legend

• Alluvial deposits
• Sandstones
• Limestones with hematite
• Sandy conglomerate
• Conglomerate
• Clay shales
• Lime–stibnite vein
• Contours of the tectonic surface of the first phase

Axonometric representation of a blocky ore vein and of the structural surface of the first phase of fracture tectonics

Fig. 2

Line V

Line IV

Line III′

Line II′

Line I′

Scales

Axonometric projection of profiles of sections of the deposit
with contour lines of the average metal contents along them.

Fig. 3

the method in central projection was considered only in the first quarter of the twentieth century, more precisely in 1910, by Erwin Kruppa, who transferred the Pohlke–Schwarz theorem to central axonometry.

In our country, questions of the theory of central axonometry began to be studied only in the Soviet period, and much that was new was introduced into the proof of Kruppa’s theorem; new theorems were also proposed.

Block diagram of a synclinal fold

Fig. 4

In 1945 N. F. Chetverukhin [17] and, simultaneously with him, N. M. Beskin [18] published their papers devoted to the proof of Kruppa’s theorem. As is known, Kruppa’s own proof, based on the properties of minimal cones and imaginary circles, was very complicated. N. F. Chetverukhin replaced the minimal cones and imaginary circles by their real “representatives,” which made it possible to subject Kruppa’s condition to graphical verification. This was achieved by using polar correspondences.

The paper by N. M. Beskin, considering Kruppa’s second theorem, proposes a somewhat refined proof, which is a natural generalization of the Pohlke–Schwarz theorem in central projection. The author conducts the proof by a projective-analytic method, as a result of which he arrives at the conclusion,

Conventional designations:
Limestone;
Calcareous shale;
Clayey shale;
Hard coal;
Sandy shale.

Block diagram of a disjunctive, or rupture, disturbance in isometric projection

Fig. 5

that, under central projection of a spatial projective coordinate system specified in any manner, one can obtain a previously specified plane image up to a unimodular-affine transformation. This conclusion fully coincides with the formulation of the Pohlke–Schwarz theorem, when, by projecting in parallel any spatial affine coordinate system (also specified in any manner), one can obtain a previously specified plane image up to similarity.

The projective-analytic proof by N. M. Beskin of Kruppa’s second theorem is not sufficiently visual and is rather complicated; therefore new proofs of this theorem were proposed. In 1949, Associate Professor E. A. Mchedlishvili, in his work [19], in search of an elementary proof, used “projective correspondences of space and flattened space or the plane of projections” (p. 175). The proof of the theorem is reduced to finding the center and the plane of projections onto which the given spatial Desargues configuration is projected into the same given plane Desargues configuration.

It should be noted that, despite the comparative brevity of the proof, the author presents it in insufficiently clear language, introducing the new concept of “flattened space,” which he regards as the “plane of the plane Desargues configuration.” E. A. Mchedlishvili himself notes that, because of the concentration of “attention on Desargues configurations, the true essence of this theorem escapes notice” (p. 178).

A more interesting and visual proof of the fundamental theorem of axonometry in central projection was achieved by I. S. Dzaparidze [20].

As was mentioned above, the theory of complete and incomplete images, which has been applied to all types of projection drawings, including those executed in central projection (perspective), is of great importance for the rigor of projection methods.

Professor O. A. Volberg [14] devoted much attention to methods for constructing a single image and, in particular, to the solution of metric problems in central projection. The principal difference between his works and all others is that he showed most fully the connection between descriptive geometry and projective geometry.

In 1951, a major work by I. I. Kotov appeared [21], constituting a monograph in which investigations of central projections of both three-dimensional and multidimensional space are presented. On the basis of an analysis of central projections, the author set the task of “developing the theory of combined images as a method for constructing plane models of spatial figures that fully reflect their positional and metric properties as figures of projective and Euclidean space” [21] (p. 5). Considering the central projection of a rectangular coordinate system onto an axonometric plane, he arrives at the conclusion that the theorem of central axonometry [17, 18] may be formulated as follows:

“The triangle of traces of the combined image of a spatial coordinate system in central projection is homothetic to the triangle of lines of convergence of its coordinate planes, with the center of homothety being the central projection of the origin of coordinates” [21] (p. 142). This theorem holds for an oblique-angled system of coordinate axes. Proceeding from the formulation of the theorem given above, the author proves the converse theorem, calling it the theorem of existence of central projections:

According to this theorem, any two homothetic acute-angled triangles having a center of homothety may be “... regarded as a combined image of a rectangular coordinate system in central projections, supplemented by projections of infinitely distant straight lines, the coordinate planes” [21] (p. 143). If the center of homothety coincides with the orthocenter of the triangle of traces, then central axonometry of this type is called normal; moreover, when two axonometric axes have congruent scales, it is normal, or central, dimetry. Normal axonometry with three congruent scales is called normal, or central, isometry. I. I. Kotov further concludes that in Old Russian miniatures a special form of normal dimetry was used (he called it “Russian dimetry”), in which the plane of axonometric projections was parallel to the axis $Z$. As for the “reverse perspective” used by Russian artists, it is only an “approximation to one of the forms of central axonometry” [21] (pp. 176–177), since in that period there could be no question of any measurements along the axes on the basis of the coordinate principle. The use of combined images of coordinate axes in central axonometry makes it possible to establish its practical use in photogrammetry and, moreover, reveals the connection between central and orthogonal axonometry, the latter being a special case of the former.

The application of central axonometry in photogrammetry is a major practical question, solving the problem inverse to the construction of perspective, i.e., the reconstruction of the original from a given photographic image.

At present the photogrammetric method is used:

a) to identify features of terrain where construction work is to be carried out;

b) for geological and military purposes (the preparation of military topographic plans);

c) in photographing architectural or archaeological objects;

d) in land reclamation, agriculture, and other fields.

For solving the inverse problem—the problem of reconstructing an object from its central image—in photogrammetry there are many methods, namely: the intersection method, the analytic method, and the graphic-mechanical method. However, none of them is based on the method of central axonometry. In a number of works [22, 23, 24, 25, etc.] this method is used to solve problems for simple geometric figures and for the special case in which, in the photograph, in addition to the images of points, their secondary projections are present (the so-called complete image in the sense of N. F. Chetverukhin). The solution of metric problems from images of objects of more complex form becomes impossible. An exception is the dissertation of V. A. Ryzhova [25], who, from two photographs of an object taken from two points, carried out their axonometrization, which made it possible to determine the axonometric coordinates of any point photographed in the terrain. The author proposed a new scheme of an apparatus for processing photographs by the coordinate method.

Axonometric projections have found extensive application in questions of physicochemical analysis, where it became necessary to use them for a multidimensional space. A corresponding contribution to the development of methods of representation in $n$-dimensional space was made by Academician N. S. Kurnakov.

“Professor N. S. Kurnakov,” wrote E. S. Fedorov, “emphasized the importance, for the purposes of physical chemistry, of a method of representation...,” which

... made images (functions) of four independent variables accessible to visual representation” [26] (p. 53).

The method for representing systems with four independent variables was proposed in 1902 by the Dutch scholar P. H. Schoute [27], who, while developing the theory of construction methods in four-dimensional space, generalized to it the fundamental theorem of axonometry. Subsequently, in 1916, Schoute’s method was applied in practice by Boeke [28], which is why the name “Schoute–Boeke method” has survived to the present day. In the character of its construction, the method mentioned may be classified as axonometric, since it is based on the coordinate principle, which is an essential condition of axonometry. Considering four variable quantities as coordinates of points in four-dimensional space, one can represent such a point on a plane in the form of a complex drawing. To do this, taking two mutually perpendicular lines, their point of intersection is taken as the origin of coordinates, and the four rays \(ox\), \(oy\), \(oz\), \(ot\) are taken as the four coordinate axes. Laying off the corresponding values of the variables \(a\), \(b\), \(c\), \(d\) and marking points on the axes \(x\), \(y\), \(z\), \(t\), one draws lines parallel to the given axes. The mutual intersection of the lines at four points \(A_1\), \(A_2\), \(A_3\), \(A_4\) represents the composition or state of the system (Fig. 6).

Fig. 6

Considering the various methods of representation in physicochemical analysis, one may note that the authors who proposed one or another method mostly used the coordinate principle for constructing “composition—property” diagrams, as a result of which many of these methods may be assigned to the method of parallel multidimensional axonometry.

The broad application of graphical methods for representing multicomponent systems received corresponding reflection in theoretical studies concerning the axonometry of \(n\)-dimensional space. At present we have works specifically considering theoretical questions of multidimensional axonometry, carried out by N. F. Chetverukhin, I. I. Kotov, Z. I. Pryanishnikova, V. N. Pervikova, and others.

It should be noted that the axonometry of multidimensional space represents a rather interesting field for further theoretical investigations and practical use.

Summing up what has been said, the following conclusions may be drawn:

1. In the Soviet period, the theory of axonometry and its practical applications received the greatest development.

2. The majority of research scholars have sought and continue to seek to reconstruct metrically a spatial object (a tetrahedron) from its image in parallel and central axonometry, i.e., to obtain the most convenient methods for proving its fundamental theorems and reconstructing the natural system of coordinates.

3. Multidimensional axonometry, which has found application in physicochemical analysis, has received corresponding development.

4. To speed up the construction of axonometric images, various designs of axonographs were proposed. It must be said that up to the present these instruments are little used in production. This is apparently explained either by the complexity of setting up the instruments (the axonograph of L. S. Skripov), or by additional graphical po-

constructions that are not used in further work (the Kohn and Krot instruments), or by the limited range of axonometric projection types that can be drawn (Yuditskii’s instrument), etc. It may be noted that, of all the designs currently in existence, the instrument of N. L. Ruskevich should be regarded as the one most suited to the level of mechanization requirements; it is fully capable of being introduced into the design and drafting offices of factories, both because of the simplicity of working with it and because of the considerable acceleration of the process of constructing images.

1. For the rapid selection of the required type of axonometric projection, the method of nomographing the parameters and formulas of orthogonal axonometry proposed by A. I. Ostrovskii and M. V. Burde is very convenient. Those interested in this method may become more fully acquainted with it in [16].

REFERENCES

1. Collection Questions of Modern Descriptive Geometry. Moscow, 1947.

2. A. K. Vlasov. A new proof of Pohlke’s theorem (Mathematical Collection, vol. 32, no. 3, 1925).

3. N. A. Glagolev. Descriptive Geometry. 1st ed., Moscow, 1936.

4. N. A. Glagolev. Generalization of Pohlke’s theorem (Mathematical Collection, vol. 32, no. 4, 1925).

5. N. F. Chetverukhin. Triples of straight lines as projections of rectangular systems of axes in space (Mathematical Collection, vol. 40, no. 4, 1933).

6. N. F. Chetverukhin. Introduction to Higher Geometry. 1st ed., Moscow, 1934.

7. N. F. Chetverukhin. On the affine rigidity of polyhedra (Reports of the Academy of Sciences of the USSR, vol. 42, no. 1, 1944).

8. N. F. Chetverukhin. Drawings of spatial figures in a geometry course. Moscow, 1946.

9. N. F. Chetverukhin. Complete and incomplete images (Questions of Modern Descriptive Geometry, Moscow, 1947).

10. N. F. Chetverukhin. Conditional images and the parametric method for constructing them (Questions of Modern Descriptive Geometry, Moscow, 1947).

11. I. M. Kipnis. Geometric problems requiring the compilation of trigonometric equations (Mathematics in School, no. 4, Moscow, 1941).

12. N. F. Chetverukhin. The fundamental theorem of axonometry from the point of view of conditional images (Proceedings of the Moscow Aviation Institute, no. VI, 1947).

13. N. F. Chetverukhin. A point basis and its application to projection drawings (Engineering Collection of the Academy of Sciences of the USSR, vol. V, no. 1, Moscow, 1948).

14. O. A. Wolberg. Lectures on Descriptive Geometry. Moscow, 1947.

15. G. Scheffers. Lehrbuch der darstellenden Geometrie. Bd. I—II. Berlin, 1919—1920.

16. E. A. Glazunov and N. F. Chetverukhin. Axonometry. Moscow, 1953.

17. N. F. Chetverukhin. On the fundamental theorem of axonometry in central projection (Report of the Academy of Sciences of the USSR, vol. L, 1945).

18. N. M. Beskin. An analogue of the Pohlke–Schwarz theorem in central axonometry (Reports of the Academy of Sciences of the USSR, vol. L, 1945).

19. E. A. Mchedlishvili. Projective Foundations of Descriptive Geometry (separate offprint from the Proceedings of TPI, no. 19). Tbilisi, 1949.

20. I. S. Dzhaparidze. A projective-synthetic proof of a proposition of N. M. Beskin (Proceedings of the Georgian Polytechnic Institute, no. 30, 1954).

21. I. I. Kotov. Combined Images. Publishing House of the Moscow Aviation Institute, 1951.

22. G. M. Deshevoi. Application of the coordinate representation method to the solution of problems in perspective (Transactions of the Leningrad Mining Institute, no. 3, 1939).

23. K. S. Kipshidze. A Method for Constructing Perspectives by Axonometric Projections. Candidate dissertation, V. I. Lenin Library, 1944.

24. A. A. Kohn. Transformations of Orthogonal Projections into Central and Axonometric Projections. Candidate dissertation, V. I. Lenin Library, 1949.

25. V. A. Ryzhova. The Method of Central Axonometry and Its Application in Photogrammetry. Candidate dissertation, V. I. Lenin Library, 1953.

26. E. S. Fedorov. Exact representation of points of space on a plane (Transactions of the Mining Institute, vol. 1, no. 1, St. Petersburg, 1907).

27. P. H. Schoute. Mehrdimensionale Geometrie. I Teil. Leipzig, 1902.

28. H. E. Boeke. Eine Anwendung mehrdimensionaler Geometrie auf chemisch-mineralogische Fragen, die Zusammensetzung des Turmalins (Neues Jahrbuch für Mineralogie, Geologie und Paleontologie. II Band. Zweites Heft. S. 109—148. Stuttgart, 1916).

Corrections and Noted Misprints

1. In an image, a spatial conjugate pair is understood to be a plane and a line to which, in the original, there correspond a mutually perpendicular plane and line. ↩