DEVELOPMENT OF E. KRUPP’S PROBLEM

V. A. Voskresenskii

(Presented by the Department of Descriptive Geometry)

Recently in the USSR much attention has been devoted to the study of the so-called problem of E. Krupp [1]. Since a number of works have appeared with various refinements of the proof of E. Krupp’s proposition, the need has arisen to make a corresponding analysis of the most interesting of them. The present article is devoted to this question.

E. Krupp’s problem concerns the fundamental proposition of axonometry, i.e., the solution of the question of the possibility for a given coordinate system \(D'\) to have a given image \(D\) in central projection.

It is known that in the case of parallel projection this problem is solved by the Pohlke–Schwarz theorem.

In central projection the question becomes more complicated, since here the invariant of the transformation is the cross ratio of four points on a line. If on the axis \(OX\) there are given points \(O, A, A_1\), corresponding respectively to the images of the points \(O', A', A'_1\), then the point \(N\), also being the central projection of the point \(N'\), can be determined from the conditions:

\[ \frac{OA_1}{A_1A} : \frac{ON}{NA} = \frac{O'A'_1}{A'_1A'} : \frac{O'N'}{N'A'} . \]

Thus, in the presence of coordinate axes with two previously indicated segments on each of them, under central projection one can obtain a plane figure specified in advance, which is, in essence, a planar Desargues configuration. Hence the fundamental problem of central axonometry is formulated by the following proposition.

If there is an arbitrary, previously given planar Desargues configuration
\[ D \equiv O(AA_1, BB_1, CC_1), \]
then it can be regarded as the central projection of an arbitrary, specified in advance, spatial Desargues configuration
\[ D' \equiv O'(A'A'_1, B'B'_1, C'C'_1), \]
and, conversely, having such a spatial configuration, it is possible to set up the central projection in such a way that we obtain a planar Desargues configuration, unimodular-affine*) to the one previously given.

Erwin Krupp in 1910 [1] solved the posed problem for central axonometry.

His first theorem states:

*) An affine transformation preserving unchanged the areas of all figures.

“For the given planar Desargues configuration to be a central projection of some isosceles rectangular tetrahedron, it is necessary and sufficient that the conic section**) \(K_i^2\) be a (imaginary) circle.”

It should be noted that in Krupp’s work a major role is played by \(K\)-systems (named after Krupp), composed of a triangle, a point, a line, and a conic section.

The proof of this theorem by Krupp himself turned out to be very complicated, since it was based on the properties of minimal cones

Fig. 1.

and imaginary circles. Therefore, in 1945 N. F. Chetverukhin [3] proposed a rather elementary proof of Krupp’s first theorem, based on polar correspondence.

However, this theorem cannot serve as a natural generalization of the Pohlke–Schwarz theorem in the case of central projection, since it imposes too many conditions, whereas the Pohlke–Schwarz theorem, without any conditions, makes it possible to obtain from the given original a projection similar to a preassigned figure.

In connection with this, Krupp’s second theorem arose:

“If an arbitrary spatial Desargues configuration \(D'\) and an arbitrary planar Desargues configuration \(D^0\) are given, then the center of projection \(S\) and the plane of projection \(\pi_0\) can be chosen so that the projection of \(D'\) is a configuration \(D\), collinear with \(D^0\).”

N. M. Beskin [4] proved this theorem by a projective-analytic method in a more precise form, assuming that the spatial Desargues configuration \(D'\) can be projected into a planar configuration \(D\) that is unimodular-affine to \(D^0\).

It should be noted that N. M. Beskin’s proof lacks sufficient visual clarity and is rather complicated; therefore, other proofs were subsequently proposed.

In 1949 Associate Professor E. A. Mchedlishvili, in his work [5], in search of an elementary proof, used “projective correspondences of space and an extended space or plane of projections.”

He formulated the fundamental theorem of central images as follows:

“If a spatial Desargues configuration is given,
\(D' \equiv O'(A'A'_1,\ B'B'_1,\ C'C'_1)\), and also a planar Desargues configuration
\(D \equiv O(AA_1,\ BB_1,\ CC_1)\), such that the triangles formed by the points under the projective correspondences of rectilinear rows \((O'A'A'_1 \wedge \overline{OAA_1});\)

**) Here the conic section is introduced on the basis of Schur’s theorem [2].

\((O'B'B'_1 \land \overline{OBB_1});\ (O'C'C'_1 \land \overline{OCC_1})\) are similar, then the given planar Desargues configuration is a central projection of a spatial Desargues configuration.”

Despite the relative brevity of the proof, the author sets it forth in insufficiently clear language, introducing the concept of a “flattened space,” which he regards as the “plane of a planar Desargues configuration.”

E. A. Mchedlishvili himself notes that, because of the concentration of “attention on Desargues configurations, the true essence of this theorem slips away” (p. 178).

On April 5, 1950, I. I. Kotov, at a meeting of the Department of Descriptive Geometry of the Moscow Aviation Institute, delivered a report “On One Form of Central Axonometry,” the materials of which were published in 1951 [6]. The author, considering the central projection of a rectangular coordinate system onto an axonometric plane, comes to the conclusion that the theorem of central axonometry can be formulated somewhat differently:

“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 intersection of its coordinate planes, with the center of homothety being the central projection of the origin of coordinates.”

To understand this theorem, the author proposes considering the following construction (Fig. 2).

In space there is a rectangular coordinate system \(O'x'y'z'\) and a plane of axonometric projections \(\pi\), intersecting the coordinate axes at the points \(x_\pi, y_\pi, z_\pi\).

If a center of projection \(S\) is given, then from it one can project the given system of axes onto the plane \(\pi\). Then we obtain a triple of straight lines \(x, y, z\), intersecting at the point \(O\) and being the projections of the axes \(x', y', z'\) and of the origin of coordinates \(O'\).

Fig. 2.

Let us supplement the resulting drawing with improper elements; for this purpose, through the center of projection \(S\) we draw planes parallel to the given coordinate planes \(x'y'\), \(x'z'\), \(y'z'\). As a result, on the projection plane we shall have the lines of intersection \(x_\infty y_\infty,\ x_\infty z_\infty,\ y_\infty z_\infty\) of the indicated coordinate planes.

Two triangles are obtained: the triangle of the vanishing lines \(x_\infty y_\infty z_\infty\) and the triangle of traces \(x'_\pi y'_\pi z'_\pi\) as the intersection of two equal trihedral right angles, \(Sx_\infty y_\infty z_\infty\) and \(O'x'y'z'\), by the plane \(\pi\). They are in fact homothetic figures with the center of homothety at the point \(O\).

Proceeding from this theorem, the author proves its converse, calling it the theorem on the existence of central projections:

“Any two homothetic acute-angled triangles with their center of homothety can always be regarded as the combined image of a rectangular coordinate system in central projections, supplemented by projections of the infinitely distant lines of the coordinate planes” (Fig. 3).

Fig. 3.

Given: the triangles \(abc\), \(def\), and the center of homothety \(O\) in the plane \(\pi\).

It is required to prove that \(od\), \(oe\), and \(of\) are central projections of the coordinate axes.

Proof. Over each triangle, on both sides of the plane \(\pi\), construct two symmetrically situated rectangular trihedral angles.

Connect the vertices of these angles to one another by the straight lines \(K_1K_3\) and \(K_2K_4\), which, being symmetric with respect to the plane \(\pi\), intersect at the point \(O\).

Consequently, the homothetic acute-angled triangles \(abc\) and \(def\), with center of homothety \(O\), determine in space four rectangular trihedral angles \(K_1abc\), \(K_2abc\), \(K_3def\), \(K_4def\), and any one of them may be taken as a coordinate trihedron whose vertex of the trihedral angle, situated on the same side of the plane \(\pi\), may be the center of projection.

If one of the homothetic triangles is regarded as the triangle of traces of a spatial coordinate system, then there will be two possible positions of it and of the center of projection.

As is known, specifying in the plane the image of a Desargues configuration
\(D \equiv (Ox'_\pi y'_\pi z'_\pi x_\infty y_\infty z_\infty)\) with an infinitely distant Desargues line determines in the plane \(\pi\) the position of the spatial coordinate system \(O'x'y'z'\) and the center of projection \(S\). In this case the problem of constructing the central axonometry of a figure from the Cartesian coordinates of its characteristic points becomes completely solvable in the given plane.

Let us consider the following example (Fig. 4).

Suppose that some point \(A'\) of the figure \(F'\) has its abscissa in the form of the segment \(O'a_{x_1}\). Superpose \(\triangle O'x'_\pi O\) and the center of projection \(S\) with the plane \(\pi\). We obtain \(\triangle O_x x'_\pi O\) and the point \(S_x\), as well as the superposed position \(O_x a'_{x'\pi}\) of the segment \(O'a_{x'}\). Thus, on the plane \(\pi\) we have the central projection of the segment \(O'a_{x'}\), represented as the segment \(Oa_x\) when projected from the superposed position of the center \(S_x\) of the segment \(O_x a'_{x'\pi}\). The construction of the superposed triangles presents no difficulty, since the true magnitudes of the segments \(OO'\) and \(x_\pi O\) are easily found

as the hypotenuses of the right triangles \(O'O_1O\) and \(x'_{\pi}O_1O'\), where \(O_1\) is the orthogonal projection of the point \(O'\) onto the plane \(\pi\).

The legs \(O_1O\) and \(x'_{\pi}O_1\) are given on the plane of the images, while \(O_1O'\) is determined as a segment that is the mean proportional to the segments \(x'_{\pi}O_1\) and \(O_1c\). \(S_x\) is determined as the intersection of the line, parallel to \(x'_{\pi}O_x\), passing through \(x_{\infty}\), with the line \(OO_x\).

I. I. Kotov then finds the perspective of a certain point in space and recommends carrying out the solution of the problem according to the scheme he developed \([6,\ p.\ 150]\).

After this he proceeds to the construction of images of any figure given by the Cartesian coordinates of its characteristic points. In order to avoid cluttering the drawing with superfluous lines, the author considers it more expedient to construct the coordinate segments \(O_{a_x}\), \(O_{a_y}\), and \(O_{a_z}\) separately, while on the principal drawing performing only the construction of the coordinate broken lines of the perspective images of the corresponding points of the figure.

Fig. 4.

Subsequently I. I. Kotov proposes imposing certain conditions, choosing the center of projection only on the line \(O_1O'\) (Fig. 4).

In this case central axonometry will have the same properties as in orthogonal axonometry (Fig. 5), i.e.

a) the origin of the coordinate system is projected inside the triangle of traces \(x'_{\pi}y'_{\pi}z'_{\pi}\), coinciding with its orthocenter;

b) the axonometric axes \(x, y, z\) will pass through the orthocenter and the vertices of the triangles of traces \(x'_{\pi}y'_{\pi}z'_{\pi}\) and of the vanishing line \(x_{\infty}y_{\infty}z_{\infty}\);

c) the angles between the axes will be only obtuse.

The author proposes calling an axonometric projection of this kind, specified by the configuration
\(D \equiv (Ox'_{\pi}y'_{\pi}z'_{\pi}x_{\infty}y_{\infty}z_{\infty})\), normal (by analogy with rectangular axonometry).

I. I. Kotov carried out investigations on the construction of central axonometry and concluded that if there are two congruent scales on the axonometric axes, then an axonometry of this kind should be called normal or central dimetry, and with three congruent scales—normal or central isometry.

The author considered in detail the question of constructing a normal central dimetry if the axonometric plane \(\pi\) is parallel to the axis \(z'\).

As an example he presented a drawing of a tower from a miniature of a Russian manuscript of the end of the fifteenth century (Fig. 6), believing that such a type of image-

… is typical for that time, and therefore it could have been called Russian dimetry.

In our opinion, such a name is highly relative, since Russian artists, chiefly icon painters, did not adhere to any metric measurements and did not construct any congruent scales.

Fig. 5.

Fig. 6.

I. I. Kotov, by his theoretical developments and their practical applications in the field of central axonometry, showed that its particular types—normal central isometry and dimetry—besides the visual clarity of the representation, possess all the advantages of orthogonal axonometric projection and, above all, sufficient simplicity and analogous methods of representation.

In this, undoubtedly, lies the great merit of Professor I. I. Kotov.

In 1954 I. S. Dzhaparidze [7] proved, by a projective-synthetic …

by way of the well-known theorem of N. M. Beskin [2, 4], and, in our opinion, this proof proves to be simpler.

Finally, in 1956 O. Ya. Ryunk [8] wrote a paper in which he noted that the most expedient generalization of Kruppa’s second theorem had been made by N. M. Beskin, but that, as a result of replacing the scale tetrahedron by the Desargues configuration, the axonometric content disappeared, since the determination of each coordinate by means of 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 aggregate of the vertices of a tetrahedron with the improper point of one scale edge. The resulting plane figure of 5 points, three of which lie on one straight line, is the central projection of such an object.

The fundamental theorem of central axonometry according to O. Ya. Ryunk was of the following content:

“The perspective of the scale tetrahedron of an orthogonal equal-scale coordinate system, together with the point of convergence of one coordinate axis, determines the position of the axes and of the center of projections relative to the picture plane.”

If we compare I. I. Kotov’s theorem with the one just cited, then a certain similarity may be observed, namely: O. Ya. Ryunk, in order to determine the position of the coordinate axes and of the center of projections (more precisely, the center of projection—V. V.), uses on the picture plane the improper point of one coordinate axis. In I. I. Kotov’s work there is mention of supplementing the projections with the improper straight lines of the coordinate planes.

It should further be noted that O. Ya. Ryunk, just like I. I. Kotov, showed a practical use of central axonometry. For this purpose he first decided to apply central axonometry analogous to orthogonal isometry.

In this case, the coordinate trihedron must be isosceles, i.e., equally inclined to the picture plane [9].

The application of the method of coordinates to the solution of problems in central projections was demonstrated in her work by S. M. Desheva [10]. She considered problems on constructing the central projection of a point in space, of a plane figure of given dimensions, and on determining the true magnitude of the angles between the principal geometric elements and the distances between them.

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

In 1959, in the works of N. V. Paluver [11, 12], certain fundamental questions of central axonometry were investigated and, in particular, the problem was considered of determining, from given images of a spatial equal-scale coordinate system with improper points on its axes, the position of the center of projection and of the scale tetrahedron relative to the plane of projections.

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

1. From the time of the appearance of E. Kruppa’s theorem there began the so-called fourth period in the development of Pohlke’s theorem. Let us recall that N. M. Beskin [2] proposed dividing the history of the improvement of the fundamental theorem of axonometry into four periods: the first—from 1853 to 1864, when its elementary proofs were absent; the second—from 1864 to 1885—the period of the generalization of the theorem by H. Schwarz

from a metric coordinate system to an affine one; the third—from 1885 to 1910, when F. Schur gave the Pohlke–Schwarz theorem a projective character; the fourth—from 1910 to the present. The works by which this period is characterized may be judged from this survey article.

As can be seen, the greatest interest in the corresponding investigations arose 35 years after the publication of E. Kruppa’s theorem. The latter is explained by the fact that by that time Chetverukhin had developed a theory of complete and incomplete images, which prepared the ground for further investigations not only in the field of parallel axonometry, but also of central axonometry.

1. Among the proofs of the fundamental proposition of axonometry, the most complete should be considered the proof by N. M. Beskin [2, 4], which refined E. Kruppa’s second theorem to a unimodular-affine correspondence.

Subsequent works—by E. A. Mchedleshvili, I. S. Dzhaparidze, I. I. Kotov, and others—come down to attempts to give the proofs graphic clarity and simplicity of explanation.

1. Analyzing the works of the above-mentioned authors, it should be noted that the most successful (in our view) are the works of I. I. Kotov [6] and I. S. Dzhaparidze [7], and in I. I. Kotov the direct application of the theory to practice is considered.

This same idea was further developed in the works of O. Ya. Ryonk [8, 9], S. M. Deshevoy [10], and others.

1. There are not so many theoretical investigations and practical applications of central axonometry, in comparison with analogous questions in orthogonal axonometric projections. The reasons for this are (in our view) the disappearance of the essence of the theorem, owing to great attention to Desargues configurations, and a certain complexity of graphic execution in solving metric problems.

2. The circumstances mentioned do not diminish the importance of the fundamental problem of central axonometry, which received its further development in multidimensional space, evidence of which may be the article by V. N. Pervikova [13].

In the above-mentioned work of I. I. Kotov [6] it is said that the investigation of the properties of central projection in multidimensional space can be reduced to the investigation of parallel projection in \(n\)-dimensional space onto a three-dimensional picture plane, with subsequent consideration of the ordinary central projection of three-dimensional space (p. 79).

1. Thus, the question of the further development of E. Kruppa’s problem will be of great importance both in the sense of improving theoretical proofs and in methods of applying them for practical purposes.

LITERATURE

1. E. Kruppa. Zur axonometrische Methode der darstellenden Geometrie (Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften. Mathematischnaturwissenschaftliche Klasse. Abt. IIa, Bd 119, Heft 4. Wien. 1910. S 487—506).

2. N. M. Beskin. The Fundamental Proposition of Axonometry (Problems of Modern Descriptive Geometry, edited by N. F. Chetverukhin, Gostekhizdat, 1947).

3. N. F. Chetverukhin. On the fundamental theorem of axonometry in central projection. (Reports of the USSR Academy of Sciences, New Series, Vol. L, 1945).

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

5. E. A. Mchedleshvili. Projective foundations of descriptive geometry. (Proceedings of the Georgian Polytechnic Institute, No. 19, Tbilisi, 1949).

6. I. I. Kotov. Combined Images. MAI, 1951.

Fig. 1. Schematic circuit diagram of the main conversion units.

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

2. O. Ya. Ryunk. The fundamental problem of axonometry in central projection. Dissertation. Tallinn, 1956.

3. O. Ya. Ryunk. On the practical application of central axonometry (Problems of the theory, applications, and methods of teaching descriptive geometry, Riga, 1960).

4. S. M. Deshevoi. Application of the method of coordinates to the solution of metric problems in central projections. (Collection of scientific works No. 13 of the Leningrad Mechanical Institute, Leningrad, 1959).

5. N. V. Paluver. An investigation of some basic questions of central axonometry. Dissertation, Tallinn, 1959.

6. N. V. Paluver. On an analytic solution of Krupp’s problem in central axonometry. (Problems of the theory, applications, and methods of teaching descriptive geometry, Riga, 1960).

7. V. N. Pevikova. A generalization of the fundamental theorem of central axonometry to an \(n\)-dimensional space (Methods of descriptive geometry and its applications, Moscow, 1955).