On the Development of Axonometry in the USSR

V. A. Voskresenskii

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

In 1967 an article by G. K. Nikolaevskii [1] appeared in print, in which a general method was proposed for carrying out axonometry of isotopic transformations of space, under which various types of one-to-one and continuous mappings preserve the position of the figure being transformed.

The author considered it possible to apply curvilinear axonometry, above all, in constructing visual representations on surfaces, recommending in this case the introduction of the term panoramic axonometry, both parallel and central.

Since it has become necessary to mention the term “parallel axonometry,” the question arises as to what was done further in this area after those achievements discussed in articles [2, 3].

In November 1955, an expanded meeting of workers of the departments of descriptive geometry and graphics of higher educational institutions of the USSR was held in Batumi, where Professor-Doctor N. F. Chetverukhin, in his report [4], described the work that was being carried out on axonometry, both central and parallel. In particular, speaking of the development of parallel axonometry, N. F. Chetverukhin mentioned the name of K. P. Voronina, a postgraduate student of the Moscow Aviation Institute.

The latter’s dissertation, published in 1957 [5], presented investigations that established simple graphical connections, in a complex drawing, between the orthogonal projections of an object and its axonometric and perspective images.

K. P. Voronina considered chiefly parallel axonometry from the standpoint of its construction. She did not deal with central axonometry, devoting attention only to perspective (central) images of an object according to its given complex projections, without carrying out additional constructions.

The works of other authors in the field of parallel axonometry concerned: a) studies of the basic formulas and theorems, and b) practical methods of construction and of solving various problems.

Subgroup “a” may include the articles of A. I. Ostrovskii [6], M. V. Burde [7, 8, 9], R. G. Shermazanov [10], and others. Among the works listed here, it is worth dwelling on the work of M. V. Burde [7], who proposed a proof of the fundamental theorem of parallel axonometry without resorting to the aid of affine correspondences, since the essence of the latter

is not considered in the courses of higher-school programs in descriptive geometry.

The author carries out the entire proof by the method of “phase transformation,” i.e., he successively rotates the image of the trihedron about its edges and thereby determines in space the direction of the projecting ray.

Interesting studies appeared in the field of nomographing axonometric projections. Among the initiators of this direction one may count A. I. Ostrovskii and M. V. Burde. The latter, in one of his articles [9], established the domains of existence of parallel axonometry for constructing nomograms.

Additional investigations of Pohlke’s theorem were carried out by R. G. Shermazanov [10], who elucidated questions concerning: a) the magnitude of the projections of three equal and mutually perpendicular segments (the edges of a tetradron); b) the position of the axonometric plane; c) the direction and angle of projection.

The source materials in solving the above-mentioned questions were the author’s own works, set forth in 1946 in his dissertation: “The Dependence between the Elements of Axonometric and Orthogonal Projection.”

Among the works of subgroup “b” the following may be noted: N. T. Chuvikova [11], I. A. Dobryakova [12], A. M. Khaskin [13], who devoted attention to producing the true form of plane figures in central axonometry, G. K. Nikolaevskii [14], R. Kh. Khorunov [15], A. G. Artyukhovskaya [16], who in her work examined the relation between central and parallel axonometry and concluded that axonometric projections are the boundary between direct and inverse perspective, A. P. Loginova [17], and others.

As can be seen from this list of authors’ names, one may judge that our scholars have devoted considerable attention to axonometric projections.

Questions of central and parallel axonometry of three-dimensional space found broad application also in multidimensional space, chiefly in representing multicomponent systems on a plane.

This was influenced by the further development of chemical technology and, in particular, glass technology, which compelled attention to be paid to improving representations of the above-mentioned systems. The continuers of this direction were L. N. Lambin and N. N. Ermolenko [18].

The method they proposed for constructing diagrams of five- and six-component systems is based on the axonometry of multidimensional simplexes, whose points represent the compositions of the systems under study.

Subsequently, in 1963, L. N. Lambin completed a Candidate dissertation in which he investigated a highly topical question—the application of the theory of completeness of images to multidimensional space.

The theory of completeness of images, created by Professor N. F. Chetverukhin, had a great influence on the solution of geometric problems in three-dimensional space, but there had been practically little work on its use in \(n\)-dimensional space.

L. N. Lambin developed in considerable detail the theory of positional and metric completeness for a space of many dimensions, proposing a number of significant theorems [19].

Using the corresponding theoretical basis, the author was able to demonstrate in practice the construction of diagrams of five-component systems based on an axonometric representation of multidimensional simplexes.

In 1959 the monograph by F. M. Perel’man [20] was published, in which methods for representing multicomponent systems were described; moreover, primary attention was given to a method of projecting four-dimensional geometric figures, representing a further development and improvement of V. P. Radishchev’s method [21].

The book describes in considerable detail the simplest four-dimensional figures used in physicochemical analysis, such as the tetradic hexahedroid \((\Pi_{8}^{6})\), the prismatic hexahedroid \((\Pi_{9}^{6})\), and the heptahedroid \((\Pi_{12}^{7})\), the pyramidal hexahedroid \((\Pi_{7}^{6})\), and the heptahedroid \((\Pi_{9}^{7})\). The letter “\(\Pi\)” (polyhedroid) is a symbol denoting a 4-dimensional figure; the lower index indicates the number of vertices, and the upper index the number of bounding three-dimensional figures.

The work of F. M. Perel’man is of interest, especially for persons concerned with the use of four-dimensional space.

In some cases of the arrangement of the elements being projected in space, axonometric projections do not always provide the required visual clarity and accuracy in solving the problem, especially for the graphical calculation of multicomponent systems.

In this connection, G. M. Deshevoi, in his work [22], established the simplest projection relations between the axonometry of four-component systems and complex projections. The problem itself, solved by the author, is of undeniable interest, but because of the poor-quality drawings explaining the text and certain misprints, it cannot always be correctly understood by the reader.

Taking into account that descriptive geometry, as a mathematical science, must be connected with analytic geometry, some researchers began to apply the methods of the latter to the construction of axonometry. In particular, M. M. Iuditskii proposed carrying out multidimensional axonometric projections in a rectangular coordinate system by the methods of analytic geometry [23].

For this purpose, first of all, the coordinates of the points of a multidimensional axonometric drawing are determined, both in central and in parallel projections, by jointly solving the equations of the projecting lines and the hyperplane.

The author derived formulas for the dependence between the distortion coefficients in oblique and rectangular axonometry constructed on a hyperplane of \(n\)-dimensional space.

In addition, M. M. Iuditskii established that the analytic method of determining the coordinates of any point of an axonometric image from the coordinates of the point of the original in a space of many dimensions, and the construction of this image in a rectangular coordinate system, make it possible to draw an \(n\)-dimensional axonometric drawing with the aid of a coordinatometrograph controlled by an electric machine.

We believe that M. M. Iuditskii’s work in the field of multidimensional axonometry, in combination with analytic geometry, is among the first to investigate this very interesting and necessary question.

We would also like to draw attention to the article by L. A. Iatskevich [24], in which the latter established the dependence between the direction of projection, the position of the axonometric axes, and the distortion coefficients, and also determined the domain of existence of orthogonal axonometry of four-dimensional Euclidean space \(E^{4}\). To obtain the necessary results, the author makes extensive use of the mathematical apparatus of analytic geometry and linear algebra, thereby once again confirming the connection of descriptive geometry with mathematics.

In his work L. A. Yatskevich uses not only domestic literature, but also foreign literature (Skout, Sommerville, Schopp). The results obtained by L. A. Yatskevich are undoubtedly of great importance for further research in the field of multidimensional axonometry.

Developing the theory of axonometric projections and using the results obtained to solve practical problems, many authors posed, and continue to pose, the question of the quickest and most convenient ways of constructing visual representations with the aid of ordinary drafting instruments, as well as by means of mechanizing and automating the execution of any type of projection.

The problem that arose in connection with the invention of various instruments for the mechanical construction of axonometric and perspective projections now consists in determining which instrument should be preferred and approved as an indispensable part of the equipment of institute drafting rooms and factory design bureaus.

One attempt to systematize and give a brief description of the principal mechanisms for drawing and transforming projections was undertaken by G. B. Valuts and G. A. Ushakov, who in 1960 published a special book on this question [25], in which they described the designs of many instruments, both domestic and foreign.

However, the authors did not provide a comparative analysis of the existing mechanisms for constructing visual representations and therefore could not give recommendations for any one model for its maximum introduction into production and into the drafting rooms of higher schools.

Consequently, this question still remains open and in the future requires new, detailed studies of all existing axonographs from the point of view of the simplicity of their design, setup, and the speed of producing images.

In 1966 a brochure by V. M. Shitov [26] appeared in print, in which transforming drafting instruments and machines, in particular axonographs, were discussed very briefly. The author noted the very simple design of an axonograph proposed by M. M. Yuditsky. The instrument consists of an ordinary and a double straightedge, which makes it possible to manufacture it with one’s own resources. V. M. Shitov considers the drawback of the instrument to be the impossibility of drawing lines in axonometry continuously, which makes it necessary to connect the separate points obtained with a ruler or a curve.

The author further noted the theoretical development of an automatic axonograph by graduate student A. F. Babushkina in 1960, who wrote the dissertation “Certain Questions in the Automation of Visual Representations.”

Among the first initiators of the idea of automating the transformation of images should be included N. D. Bagrationi [29], who presented a report on this question at the Riga Conference on Descriptive Geometry in 1957. The author proposed an electrical system for reading a drawing, with the aid of which an axonometric image can be obtained. In this case, depending on the arrangement of the deflecting coils of the receiving tube, oblique and rectangular axonometric projections are obtained. In this case we shall have only a visual image on the screen. Consequently, this type of axonometry can be used to explain a complex composite drawing.

In order to construct the required axonometry from given orthogonal projections, one should have not an electrical system for reading the latter, but an axonograph which, unlike mechanical designs, possessed the capability of automatic execution. Such an “auto-axonograph” was proposed by N. L. Russkevich [27]. This instrument the author intended—

is used to replace the manual labor of mining surveyor engineers, architects, designers, etc., employed in drawing axonometric views. The chief advantage of the proposed construction consists in the complete automation of obtaining the original image and drawing the axonometry of plane figures.

The drawbacks of the instrument, in our view, include the complexity of constructing axonometric projections of solids, since for this it is necessary to draw a number of plane parallel sections of the given solid and then assemble the spatial image.

N. L. Russkevich’s auto-axonograph is a further practical embodiment of the idea of the automatic production of axonometric projections, the beginning of which was laid by A. V. Dmitrevsky’s construction of an electric axonograph [28].

Having considered the above-mentioned works, the following conclusions may be drawn:

1. One of the first researchers to apply curvilinear axonometry in practice may be considered G. K. Nikolaevsky.

2. In the field of parallel axonometry, mention should be made of the article by M. V. Burde [8], who proposed a simpler proof of Pohlke’s theorem, without using affine correspondences.

3. The work of L. N. Lambin deserves attention; in it the completeness of images of multidimensional objects was established [19].

4. One of the first monographs on the classification of images of multicomponent systems may be considered the book by F. M. Perelman [20].

5. The number of scientific research works on descriptive geometry in general, and on axonometry in particular, carried out in contact with mathematics, has increased noticeably, as is evidenced by the articles of L. A. Yatskevich [24], G. K. Nikolaevsky [14], and others.

6. In the field of producing axonometric projections with the aid of instruments, a significant step forward was made, characterized by the creation of electro-, auto-, and video-axonographs (A. V. Dmitrevsky [28], N. L. Russkevich [27], N. D. Bagrationi [29], and others).

REFERENCES

1. G. K. Nikolaevsky. Axonometry of isotropic transformations of space. Publishing House of Saratov University, 1957.

2. V. A. Voskresensky. A brief outline of the development of axonometry in the Soviet Union. In: Questions of the Theory, Applications, and Methods of Teaching Descriptive Geometry. Riga. Publishing House of the Riga Institute of Civil Aviation Engineers, 1960.

3. V. A. Voskresensky. On theoretical studies in the field of axonometry. Publishing House of TGU, Izv. TPI, vol. 143, 1966.

4. N. F. Chetverukhin. On scientific research work in the field of descriptive geometry. Transactions of the Georgian Polytechnic Institute. Collection “Descriptive Geometry and Engineering Graphics,” No. 1 (49), Tbilisi. Publishing House of the Georgian Polytechnic Institute, 1957.

5. K. P. Voronina. Properties of projection fields and their application to the construction of images. Proceedings of the Voronezh State Pedagogical Institute, vol. XXIV, 1957.

6. A. I. Ostrovsky. Basic formulas of parallel axonometry. Collected Works of the Moscow Seminar on Descriptive Geometry and Engineering Graphics. Moscow, Gostekhizdat, 1958.

7. M. V. Burde. Reconstruction of a scale trihedron from its parallel projection by the phase-transformation method. Collected Works of the Moscow Seminar on Descriptive Geometry and Engineering Graphics. Moscow, “Soviet Science,” 1958.

1. M. V. Burde. A new proof of certain theorems of rectangular axonometry. In: Proceedings of the Moscow Seminar on Descriptive Geometry and Engineering Graphics. Moscow, “Sovetskaya nauka,” 1958.

2. M. V. Burde. Domains of existence of parallel axonometry as a basis for constructing nomograms. In: Proceedings of the Moscow Seminar on Descriptive Geometry and Engineering Graphics. Moscow, “Sovetskaya nauka,” 1958.

3. R. G. Shermazanov. On Pohlke’s theorem. Proceedings of the Georgian Polytechnic Institute. Collection “Descriptive Geometry and Engineering Graphics,” No. 1 (49), Tbilisi, Publishing House of the Georgian Polytechnic Institute, 1957.

4. N. T. Chuvikov. Transformation of Orthogonal Projections. Moscow, “Sovetskaya nauka,” 1957.

5. I. A. Dobryakova. On the use of auxiliary axonometric projections. In: Proceedings of the Moscow Seminar on Descriptive Geometry and Engineering Graphics. Moscow, “Sovetskaya nauka,” 1958.

6. A. M. Khaskin. Certain questions in the study of axonometric projections. Proceedings of the Ukrainian Conference on Problems of Descriptive Geometry and Engineering Graphics. Kiev, 1958.

7. G. K. Nikolaevsky. On one variety of the method of two parallel projections. In: Engineering Graphics, issue I. Kharkov, Publishing House of Kharkov University, 1963.

8. R. Kh. Khorunov. On the problem of constructing plane sections of bodies in axonometry. Problems of Descriptive Geometry and Engineering Graphics, issue 39, Publishing House of the Academy of Sciences of the Uzbek SSR, 1966.

9. A. G. Aryukhovskaya. Construction of a series of related visual representations in central and parallel axonometry. In: Problems of the Theory, Applications, and Methods of Teaching Descriptive Geometry. Riga, Publishing House of the Riga Institute of Civil Air Fleet Engineers, 1960.

10. A. P. Loginov. Optimal values of the radii \(r\) and \(R\) of arcs of oval circles in constructing a circle in axonometry. In: Proceedings of the Volga Council, issue I, Saratov, Publishing House of Saratov University, 1967.

11. L. N. Lambin, N. N. Ermolenko. A method for representing multicomponent systems on a plane. Proceedings of the Ukrainian Conference on Problems of Descriptive Geometry and Engineering Graphics. Kiev, Publishing House of the Ministry of Higher Education of the Ukrainian SSR, 1958.

12. L. N. Lambin. A theory of the completeness of representation of multidimensional objects. Proceedings of the Moscow Scientific-Methodological Seminar on Descriptive Geometry and Engineering Graphics, issue II. Moscow, 1963.

13. F. M. Perelman. Methods for Representing Multicomponent Systems. Moscow, Publishing House of the Academy of Sciences of the USSR, 1959.

14. V. P. Radishchev. On methods of representation used in physicochemical analysis. In: N. S. Kurnakov, Introduction to Physicochemical Analysis, 4th ed., Publishing House of the Academy of Sciences of the USSR, 1940.

15. G. M. Deshevoy. Application of the method of projections with numerical elevations to the solution of positional problems in axonometry. In: Proceedings of the Leningrad Mechanical Institute, issue No. 21. Leningrad, Publishing House of the Leningrad Mechanical Institute, 1961.

16. M. M. Yuditsky. Construction of a multidimensional axonometric drawing in a rectangular coordinate system by methods of analytic geometry. Abstracts of Reports of the Second Scientific-Methodological Conference on Descriptive Geometry and Engineering Graphics of Higher Educational Institutions of the Uzbek SSR with the Participation of Representatives of Higher Educational Institutions of the Fraternal Republics. Tashkent, Publishing House of the Tashkent Institute of Railway Transport Engineers, 1965.

17. L. A. Yatskevich. Fundamental propositions of axonometry of \(n\)-dimensional Euclidean space \(E^n\). Problems of Applied Geometry. Collection of works by postgraduate students and applicants. Moscow, Publishing House of the Moscow Aviation Institute, 1966.

18. G. B. Valts, G. A. Ushakov. Mechanisms for Drawing and Transforming Projections. Kharkov, Publishing House of Kharkov University, 1960.

19. V. M. Shitov. Mechanization of Drafting and Design Work. Moscow, “Mashinostroenie,” 1966.

1. N. L. Russkevich. Automatic photoelectric axonograph. In: Problems of the Theory, Applications, and Methods of Teaching Descriptive Geometry. Riga: Publishing House of the Riga Institute of Civil Aviation Engineers, 1960.

2. A. V. Dmitrevsky. Theory and design of an electric axonograph. Proceedings of the Georgian Polytechnic Institute. Collection on Descriptive Geometry and Engineering Graphics, No. 1 (49), Tbilisi: Publishing House of the Georgian Polytechnic Institute, 1957.

3. N. D. Bagrationi. The idea of automating image transformation. In: Problems of the Theory, Applications, and Methods of Teaching Descriptive Geometry. Riga: Publishing House of the Riga Institute of Civil Aviation Engineers, 1960.