ON THE MAIN DIRECTIONS OF DEVELOPMENT OF AXONOMETRIC PROJECTIONS IN THE USSR

V. A. VOSKRESENSKII

(Presented by the scientific seminar of the departments of descriptive geometry of the Tomsk higher educational institutions)

Axonometric projections, being one of the methods of representation in descriptive geometry, have found wide application in geology, mining and civil engineering, theoretical mechanics, shipbuilding, physico-chemical analysis, aircraft construction, photogrammetry, and other branches of science and technology.

This spread of axonometry is explained by its advantages over complex drawings¹: sufficient visual clarity and convenience of construction by the coordinate principle.

The questions of the development of axonometric projections in our country have not, up to the present time, received comprehensive and thorough scientific study. The brief information on the history of axonometry that can be found in the works of N. A. Rynin [2], Yu. K. Vakhtin [3], N. M. Beskin [4], V. N. Bulanov [5], D. P. Volkov [6], and others constitutes a chronological list of the principal works on the axonometric method and does not give an adequate idea of the successes achieved.

Axonometry entered the course of descriptive geometry, above all, as a method of parallel projection of figures onto one plane. Therefore, for a long time it was erroneously believed that pictorial representations executed in parallel projection could be axonometric. Meanwhile, on the basis of the definitions of E. Müller [7, pp. 166–172], N. M. Beskin [4, pp. 56–57], and N. F. Chetverukhin [8, pp. 54–55], it has been established that a drawing of the original will be axonometric only in the case when it is executed according to the coordinate principle, according to which the points of the object are obtained by successively laying off their coordinates, at the corresponding scale, under any method of projection (parallel or central).

The beginning of central axonometry was laid in 1910 by the German scholar E. Kruppa, who transferred the fundamental theorem of parallel axonometry (the Pohlke–Schwarz theorem) to central projection. However, in the pre-revolutionary period in our country, due attention was not paid to questions of research into central axonometry.

In the Soviet period, the rapid development of science and technology, with highly complex instruments and machines, and the construction of large architectural and engineering structures, required the use of a large number of visual and conveniently measurable representations, executed both in parallel and in central axonometry, which in turn gave impetus to new theoretical investigations in this field.

The present article gives a brief account of three principal directions in the field of axonometric projections, established by the author as a result of a scientific analysis and systematization of a whole series of works by Soviet scholars.

The first direction should include works concerning new proofs and supplements to the fundamental theorem of axonometry in parallel projection. The second comprises works devoted to the improvement of E. Kruppa’s theorem. Finally, to the last we assign the works of our scholars in the theory and practice of axonometry for multidimensional space. It should be noted that no sharp boundary can be drawn between the first two and the last directions, since the theorems of multidimensional axonometry are also based on the parallel and central methods of projection.

Let us briefly consider what is noteworthy about the three principal directions mentioned above in the development of axonometric projections.

First of all, the beginning of investigations in parallel axonometry was provided by the lectures of Professor A. K. Vlasov of Moscow University [9], who, using the method of affine correspondence, gave a new proof of Pohlke’s theorem. This theorem received further development in the works of N. A. Glagolev [10], N. F. Chetverukhin [11, 12], N. M. Beskin [4], and others.

A major contribution to the science of projective representations was made by Professor N. F. Chetverukhin [13], who developed a new theory of complete, incomplete, and conditional representations.

This theory arose from the practice of using visual representations to explain various problems, theorems, etc., both in the process of teaching in secondary schools and institutes and in making sketches of technical parts in the design and construction bureaus of industrial enterprises.

Independently of N. F. Chetverukhin, the Leningrad professor O. A. Volberg [14] was interested in this question; he did not manage to complete his investigations and published them in the form of separate chapters devoted to the theory of one representation (monoprojection).

Questions of improving the proofs of the Pohlke–Schwarz theorem, as well as practical methods for solving metric problems, were widely reflected in a whole series of works by scholars of Soviet higher education, such as B. N. Nikolaev [15], V. A. Zelenin [16], and many others. To these should also be added the works of L. S. Skripov [17], A. M. Krot [18], G. P. Vyatkin [19], N. L. Russkevich [20], and others, concerning the mechanization of the construction of parallel axonometric projections.

Interesting investigations were carried out by N. F. Chetverukhin [21], I. I. Kotov [22], I. S. Dzhaparidze [23], A. A. Konom [24], V. A. Ryzhov [25], and others on the proof of the fundamental theorem of axonometry under central projection, as well as on its practical use in solving problems of photogrammetry.

However, despite the existing achievements in the field of central axonometry, there still remain many unresolved problems, such as

for example, the study of the locus of centers of projection of a given tetrahedron into a given quadrilateral, as well as the search for the simplest methods of constructing central axonometric images [26].

The next major direction in the development of the theory of axonometry is the work carried out by our scholars in the field of axonometry of multidimensional space, which arose as a result of requirements, in particular, of physicochemical analysis. The latter was the cause that compelled the development of methods for the graphical representation of various “composition–property” diagrams of multicomponent systems.

It is clear that the construction of diagrams of two- and three-component systems caused no difficulties, since such diagrams can be constructed with sufficient clarity in a system of three dimensions. But when it became necessary to represent multicomponent systems graphically, one had to pass to a space of four or more dimensions.

Considering the various methods of representation in physicochemical analysis, it may be noted that their authors used the coordinate principle for constructing “composition–property” diagrams; as a result, many of these methods, such as, for example, Skautte-Bouquet’s, V. Ya. Anosov’s spiral coordinates, V. N. Nikolaev’s sixty-degree coordinates, and others, can be assigned to the method of parallel axonometry in multidimensional space, although none of the authors called the mentioned methods axonometric.

Thus, a large body of research work in discovering the simplest and most convenient methods of representation in \(n\)-dimensional space belongs to our chemists, in particular to the school of physicochemical analysis of Academician N. S. Kurnakov.

Graphical methods for representing multicomponent systems have found reflection in theoretical studies of multidimensional axonometry.

At present we have works on this question by N. F. Chetverukhin [27], V. N. Pervikova [28], Z. I. Prinishnikova [29], and others, devoted to proofs of the fundamental theorem of axonometry for multidimensional space under parallel and central methods of projection.

These studies show what profound significance axonometric projections have and what their principal directions are in the matter of the further development of this method.

LITERATURE

1. Chetverukhin N. F. et al. A Course in Descriptive Geometry, Moscow, 1956.

2. Rynin N. A. Axonometry, Petrograd, 1922.

3. Vakhtin Yu. K. The Russian School of Engineering Graphics, Moscow, 1952.

4. Beskin N. M. The basic proposition of axonometry, in Questions of Contemporary Descriptive Geometry, Gostekhizdat, 1947.

5. Bulanov V. N. A Study of the Development of Engineering Graphics in Russia up to the Nineteenth Century, candidate dissertation, V. I. Lenin Library, Moscow, 1954.

6. Volkov D. P. An Essay on the History of the Development in the Nineteenth Century of a Course of Complex Projections in Russia, candidate dissertation, V. I. Lenin Library, Moscow, 1950.

7. Müller E. Vorlesungen über darstellende Geometrie, Bd. I, Leipzig und Wien, 1923.

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

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

1. N. A. Glagolev. Generalization of Pohlke’s theorem, Matematicheskii sbornik, vol. 32, no. 3, 1925.

2. N. F. Chetverukhin. Triples of straight lines as projections of rectangular systems of axes in space, Matematicheskii sbornik, vol. 40, no. 4, 1933.

3. N. F. Chetverukhin. The fundamental theorem of axonometry from the point of view of conditional images, Transactions of the Moscow Aviation Institute, issue 6, MAI Publishing House, 1947.

4. N. F. Chetverukhin. Drawings of spatial figures in the geometry course, Moscow, 1946.

5. O. A. Volberg. Lectures on descriptive geometry, Uchpedgiz, 1947.

6. B. N. Nikolaev. A new proof of Pohlke’s theorem, collection of the Leningrad Institute of Railway Engineers, issue 101, 1929.

7. V. A. Zelenin. Theory of general axonometry, Proceedings of the Ivanovo-Voznesensk Polytechnic Institute, vol. IX, 1926.

8. L. S. Skripov. Axonograph, candidate’s dissertation, Library of TPI, Tomsk, 1940.

9. A. M. Krot. On the question of the theory and design of mechanical instruments for transforming projections, candidate’s dissertation, V. I. Lenin Library, 1952.

10. G. P. Vyatkin. Mechanisms for drawing plane curves and axonometric images, candidate’s dissertation, V. I. Lenin Library, 1949.

11. N. L. Ruskevich. Comparison of instruments for the mechanical drawing of perspective and axonometry and their design on the basis of new rational graphic methods, candidate’s dissertation, V. I. Lenin Library, 1952.

12. N. F. Chetverukhin. On the fundamental theorem of axonometry in central projection, Reports of the Academy of Sciences of the USSR, vol. L, 1945.

13. I. I. Kotov. Combined images, collection Investigations on the Foundations of Descriptive Geometry, MAI Publishing House, 1951.

14. I. S. Djaparidze. Projective-synthetic proof of the proposition of N. M. Beskin, Transactions of the Georgian Polytechnic Institute, no. 30, Tbilisi, 1954.

15. A. A. Kon. Transformation of orthogonal projections into central and axonometric projections, candidate’s dissertation, V. I. Lenin Library, 1949.

16. V. A. Ryzhova. The method of central axonometry and its application in photogrammetry, candidate’s dissertation, V. I. Lenin Library, 1953.

17. N. F. Chetverukhin. On research work in the field of descriptive geometry, Transactions of the Georgian Polytechnic Institute, collection on descriptive geometry and engineering graphics, no. 1 (49), pp. 10, 11, Tbilisi, 1957.

18. N. F. Chetverukhin. Über ein axonometrisches Problem im mehrdimensionalen Raume (On an axonometric problem in multidimensional space), Matematicheskii sbornik, New Series, vol. 1/43, no. 2, Publishing House of the Academy of Sciences of the USSR, 1936.

19. V. N. Pervikova. Generalization of the fundamental theorem of central axonometry to a space of \(n\) dimensions, collection of articles Methods of Descriptive Geometry and Its Applications, edited by N. F. Chetverukhin, Gostekhizdat, Moscow, 1955.

20. Z. I. Pryanishnikova. Investigation of the metric definiteness (reversibility) of projection images and their application, candidate’s dissertation, V. I. Lenin Library, 1954.

1. A complex drawing is the name given to several mutually related projections of the represented figure [1]. ↩