ON SOME WORKS IN THE FIELD OF AXONOMETRY

V. A. Voskresenskii

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

This article briefly considers the works on axonometry which, in our opinion, are the most interesting among those carried out by Soviet scholars.

It is known that in the theory of axonometric projections, which is of interest both to mathematicians and to engineering and technical workers, there is the question of the so-called “Kruppa problem,” consisting in solving the problem of obtaining, under central projection, the image of a given coordinate system. Under parallel projection this problem is solved comparatively simply by the Pohlke–Schwarz theorem.

But under central projection the problem becomes more complicated, since here the invariant of the transformation is the cross ratio of four points on a line. As a result, we have to deal with Desargues configurations. There are several proofs of the fundamental theorem of axonometry in the case of central projection, as discussed in article [1].

Further investigations of the above-mentioned problem were carried out by O. Ya. Ryunk in his candidate’s dissertation on the topic “The Fundamental Problem of Axonometry in Central Projection,” completed in 1956 in Tallinn, where he noted that the most appropriate generalization of Kruppa’s second theorem had been made by N. M. Beskin, but that, as a consequence of replacing the scale tetrahedron by a Desargues configuration, the axonometric content disappeared, since the determination of each coordinate through a double ratio is connected with a very large number of computations, which is unsuitable for practical purposes.

In connection with this, the question arose of choosing such a figure which, in form, would be close to a tetrahedron. O. Ya. Ryunk proposed that such a figure be taken to be the set of vertices of a tetrahedron together with the improper point of one scale edge. The resulting plane figure of 5 points, three of which lie on one line, is the central projection of such an object.

The fundamental theorem of central axonometry, according to O. Ya. Ryunk, is formulated as follows: “The perspective of the scale tetrahedron of an orthogonal equiscale coordinate system, together with the vanishing point of one coordinate axis, determines the position of the axes and of the center of projection relative to the picture plane.”

Here it is appropriate to recall the formulation of this theorem by Professor I. I. Kotov [2]: “The triangle of traces of the combined image-

…of a spatial coordinate system in central projection is homothetic to the triangle of the lines of convergence of its coordinate planes, with the center of homothety being the central projection of the origin of coordinates.”

If one compares I. I. Kotov’s proposition with the formulation of O. Ya. Ryunk, one can note a certain similarity, namely: in order to determine the position of the coordinate axes and the center of projection (more precisely, the center of projecting) on the picture plane, O. Ya. Ryunk uses the improper point of one coordinate axis. Professor I. I. Kotov speaks of supplementing the coordinate planes by projections of improper lines, which will correspond to three improper points on the coordinate axes.

It should further be noted that O. Ya. Ryunk, like Professor I. I. Kotov, demonstrates the practical use of central axonometry analogously to orthogonal isometry. In this case the coordinate trihedron must be isoclinal, i.e., equally inclined to the picture plane [3].

As is known, the main metric problem of central axonometry reduces to the construction of a plane Desargues configuration representing the projection of a previously given spatial Desargues configuration. Here it is appropriate to note that V. N. Pervikova, in her article [4], considered a special case of this problem, when the spatial Desargues configuration is a tetrahedron supplemented by the improper elements of its edges and faces. The author proved theorems on determining the position of the center of projecting and the projection plane for a tetrahedron both of arbitrary form and with a right trihedral angle.

S. M. Deshevaya demonstrated the application of the coordinate method to solving problems in central projections in her work [5]. She considered problems involving the construction of the central projection of a point in space, of a plane figure of given dimensions, and the determination of the true magnitudes of angles between the basic geometric elements and of the distances between them.

The difficulty of carrying out such examples lies in the complexity of the constructions, when a whole series of auxiliary graphical operations is superimposed on the main drawing.

An interesting review of works connected with central axonometry was made by Associate Professor V. A. Manevich in 1960 in a report at a scientific conference of the departments of descriptive geometry of the Moscow Aviation Institute [6]. In this communication the author noted the dissertation of O. Ya. Ryunk and the article by N. V. Paluver [7]. The remarks expressed by V. A. Manevich concerned above all the investigations of O. Ya. Ryunk, who reduced all solutions of one variant of central axonometry to a system of equations of higher degrees, without solving its compatibility. As for N. V. Paluver’s work, here the expositions given in O. Ya. Ryunk’s dissertation were somewhat simplified.

V. A. Manevich proposed a constructive, purely synthetic solution of the problem of the possibility of centrally projecting a given arbitrary tetrahedron into a given complete quadrilateral on the basis of a central correspondence.

N. F. Chetverukhin, in his article [8], published in 1963, noted that by that time rather significant successes had been achieved in our country in connection with various generalizations of the main theorem of central axonometry. These include the above-mentioned investigations of I. I. Kotov, O. Ya. Ryunk, N. V. Paluver, V. A. Mane-

...vich and others. In addition, V. N. Pervykova determined that the locus of the centers of projection of an arbitrary tetrahedron is a spatial curve of the 16th order.

On this basis it became possible to draw some conclusions about the projection of a given tetrahedron with a point on an edge into a given plane quadrilateral with a point on its side. N. F. Chetverukhin proposed this problem in November 1955 at a conference of departments of descriptive geometry in Batumi [9]. As a result, in the general case the problem received about 8 solutions, but it is not excluded that all of them are imaginary, and thus the question of the existence of a real projection remains open, although Professor I. I. Kotov established that in certain cases the problem has a real solution. In addition, he considered the problem of projecting a tetrahedron with a point on a face, but he was able to obtain a positive solution only under the condition that the point lies on a curve of the 8th order.

N. F. Chetverukhin noted that “attempts to consider and solve these questions in the foreign literature have produced nothing new.”

Speaking of the multidimensional generalization of the main theorem both in our country and abroad, N. F. Chetverukhin named V. N. Pervykova and the Czechoslovak geometer Václav Havel, who gave multidimensional generalizations of the theorems of Pohlke, Gauss, Weisbach, and Stiefel under the condition of projection onto a hyperplane. In addition, many results of a particular character have been obtained by Soviet and foreign scholars (Kotov, Lambin, Stiefel, Hohenberg, and others).

N. M. Beskin made a major contribution to the development of the theory of axonometry. He examined in considerable detail the interrelations among various existence theorems in central axonometric projections. He was perhaps the first in the history of the development of axonometry to generalize the theorems of Ceva and Menelaus to \(n\)-dimensional space, which yielded additional results later used for solving the basic problems of central axonometry. Previously N. M. Beskin had shown how the earlier published investigations [10, 11] in three-dimensional space obtained a foundation for consideration of \(n\)-dimensional space.

The author synthetically proved new theorems of central axonometry in non-Euclidean space. Although the results obtained have not yet found direct application, nevertheless they may serve in the future for solving various practical problems, especially in constructing axonometric projections according to prescribed requirements.

A great merit of N. M. Beskin should be considered his treatment of three basic problems of central axonometry, whose essence is as follows:

A) What are the possibilities for determining the projecting apparatus (the center of projection and the plane of projection) in order to obtain an image \(D\), congruent to \(D^{0}\), if an arbitrary spatial Desargues configuration \(D'\) is given \((O'; A'_1, A'_2, A'_3; B'_1, B'_2, B'_3; C'_1, C'_2, C'_3)\) and an arbitrary plane Desargues configuration \(D^{0}\) \((O^{0}; A^{0}_1; A^{0}_2; A^{0}_3; B^{0}_1, B^{0}_2, B^{0}_3; C^{0}_1, C^{0}_2, C^{0}_3)\).

B) With what minimal distortion can the given model be reproduced, if in the general case it is impossible to obtain an image \(D\) congruent to the given model \(D^{0}\)?

C) Under what conditions imposed on the pair \((D'D^{0})\) can one obtain an image \(D\) congruent to the given model \(D^{0}\)?

N. M. Beskin does not miss the opportunity to make a certain historical excursus concerning the development of the fundamental theorem of axonometric projections. Speaking of the origin of the fundamental theorem of parallel axonometry, he noted that the original discovery of this theorem should belong to C. F. Gauss (1777–1855), the German mathematician, who in 1831 considered a particular case of the orthogonal projection of three mutually perpendicular straight lines.

But since Gauss presented this theorem in order to illustrate the use of complex numbers in geometry, he did not attach to it significance as a theorem concerning axonometric parallel projections.

In his doctoral dissertation, “Theoretical Foundations of Multidimensional Central Axonometry,” completed in 1967, N. M. Beskin gave a more exact translation of Kruppa’s second theorem, which is formulated as follows: “Three arbitrary pairs of points \(x'' x''_f;\ y'' y''_f;\ z'' z''_f\) in the image plane, whose carriers intersect at one point \(u''\), may be regarded as the secondary image of a trihedron \(u(x,y,z)\), congruent to a previously given trihedron, where \(x''_f,\ y''_f,\ z''_f\) are the vanishing points of the axes” (\(f\) is the initial letter of the word Fluchtpunkt).

In addition, N. M. Beskin noted that questions of central axonometry interested L. Hofmann, who in 1926 proved Kruppa’s first theorem for the degenerate case, when in the given configuration \(O'A'B'C'\) we have \(\angle A'O'B' = \angle B'O'C' = \angle C'O'A' = 90^\circ\), and \(O'A' = O'B' = O'C'\), while the points \(A'\), \(B'\), and \(C'\) are improper.

The same theorem, independently of L. Hofmann, was proved by E. M. Mchedlishvili, but for the nondegenerate case.

In connection with this N. M. Beskin proposes calling the theorem relating to the nondegenerate case the theorem of E. M. Mchedlishvili, and that relating to the degenerate case the theorem of L. Hofmann.

Continuing further investigations, N. M. Beskin applied these theorems to \(n\)-dimensional space and published the results in the Transactions of the Georgian Polytechnic Institute in 1967 [12].

N. M. Beskin’s works in the field of axonometry are of corresponding interest for geometers and mathematicians, since in most cases the author presents proofs of various theorems by a purely mathematical route, making little use of graphical representations.

It should be noted that problem A), that is, the determination of the projecting apparatus, was taken up by L. V. Borodin, who formulated two existence theorems for central projections [13].

Somewhat apart, in comparison with all the existing works concerning axonometric projections, stands the monograph of L. N. Likhachev [14]. The problems posed by him belong to the modern theory of images and are connected with the application of projective and projection methods to the kinematic analysis of the laws of motion of objects from motion-picture films depicting this motion. The theory proposed by L. N. Likhachev, the so-called kineperspective, synthesizes questions of projective geometry, cinema, parallel projection, kinematics, and central axonometry.

The author had previously considered the perspective-orthogonal conjugations he had composed (POC projections). Relying on the well-known work of N. A. Rynin [15], L. N. Likhachev investigated the motion of photographed objects on the basis of a series of their successive images in motion pictures.

He devoted great attention to the metric analysis of the latter, and also to the accuracy of the results obtained. Thus, the work of L. N. Li-

Khachev’s, despite the specificity of the questions he considered, may be classified among works that solve metric problems under central projection. In the author’s opinion, the methods of cineperspective analysis provide greater possibilities for its practical application not only to motion pictures, but also to the solution of certain dynamic problems of mechanics, which should subsequently constitute a new field of research, called dynamoperspective.

In conclusion to this article, the following conclusions may be drawn:

1. The study of the “Kruppa problem” is one of the important questions in the theory of modern axonometric projections.

2. The most interesting solutions of the stated problem in central axonometry were given by our scholars: I. I. Kotov, O. Ya. Ryunk, V. N. Pervikova, V. A. Manevich, N. M. Beskin, L. N. Likhachev, and others.

REFERENCES

1. V. A. Voskresenskii. On theoretical studies in the field of axonometry. Publ. Tomsk State Univ., Izv. TPI, vol. 143, 1966.

2. I. I. Kotov. Combined Images. Moscow: Publ. Moscow Aviation Institute, 1951.

3. O. Ya. Ryunk. On the application of central axonometry in practice. In: Questions of the Theory, Applications, and Methods of Teaching Descriptive Geometry. Riga: Publ. Riga Institute of Civil Aviation Engineers, 1960.

4. V. N. Pervikova. The basic metric problem of central axonometry. In: Questions of the Theory, Applications, and Methods of Teaching Descriptive Geometry. Riga: Publ. Riga Institute of Civil Aviation Engineers, 1960.

5. S. M. Deshevoi. Application of the coordinate method to the solution of metric problems in central projections. In: Collection of Scientific Works of the Leningrad Mechanical Institute, issue 13. Publ. Leningrad Mechanical Institute, 1959.

6. V. A. Manevich. Review of several works connected with the fundamental theorem of central axonometry. Abstracts of reports of the scientific conference of the Department of Descriptive Geometry. Publ. Moscow Aviation Institute, 1960.

7. N. V. Paluver. Determination of a system of coordinate axes in space from its central projection. Transactions of the Tallinn Polytechnic Institute, Series A, No. 100, Tallinn, 1957.

8. N. F. Chetverukhin. On the directions and organization of scientific work at departments of descriptive geometry and drawing. Proceedings of the Moscow Scientific-Methodological Seminar on Descriptive Geometry and Engineering Graphics, issue II, Moscow, 1963.

9. 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: Publ. Georgian Polytechnic Institute, 1957.

10. 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.

11. N. M. Beskin. Methods of representation. Encyclopedia of Elementary Mathematics, book 4, Geometry. Moscow: Fizmatgiz, 1963.

12. N. M. Beskin. The theorems of Hoffman and Mchedlishvili in \(n\)-dimensional space. Transactions of the Georgian Polytechnic Institute. Collection of works on descriptive geometry and engineering graphics. Tbilisi: Publ. Georgian Polytechnic Institute, 1967.

13. L. V. Borodin. On the question of metrization of central projections. Materials for the scientific session on applied mathematics and mechanics. Novokuznetsk: Publ. Siberian Metallurgical Institute, 1967.

14. L. I. Likhachev. Cineperspective. Riga: Publ. Riga Institute of Civil Aviation Engineers, 1960.

15. N. A. Rynin. Cineperspective. Moscow: Kinofotoizdat, 1936.