SOME ISSUES IN THE HISTORY, CLASSIFICATION, AND ANALYSIS OF THE THEORETICAL ACCURACY OF SPATIAL MECHANISMS FOR GENERATING ELLIPSES

E. I. TOROPYGIN

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

Our designers and draftsmen do not use mechanisms for drawing ellipses in their practice because of the absence of their serial production and their weak popularization. Descriptions of such mechanisms and devices are found only in the patent literature and in special journals and collections published in very limited editions. The two known works in the domestic literature devoted to questions of the theory and description of drafting mechanisms [1, 2] are now very outdated.

A substantial impediment to the broad application of the indicated mechanisms in drafting practice is also the absence of data on the results of their use, since by no means all mechanisms known, for example, from the patent literature correspond to their direct purpose.

The present article, along with questions of the history and classification of mechanisms for generating ellipses, sets forth the results of an analysis of the theoretical accuracy of operation of a certain group of the mechanisms under consideration.

History and State of the Question

Work on the creation of mechanisms for drawing ellipses has its origin in remote antiquity. Leonardo da Vinci already invented an ellipsograph for machining elliptical surfaces, which could also be used for drawing ellipses. At the end of the last century such mechanisms were known as Tülmin’s ellipsograph [1], Delone’s mechanism for drawing ellipses [2], and a number of others. At present more than 80 designs of mechanisms for drawing ellipses are known to us, not counting more complex ones, such as axonographs, affinographs, conicographs, which in particular cases can also draw ellipses.

However, despite such a comparatively long existence of mechanisms for drawing ellipses, questions of the theory of con-

...to their design, to the investigation of the theoretical accuracy of tracing curves, and to the development of criteria for assessing the quality of their operation under various conditions. It is known only that Delone, in his book [2], setting forth questions of the theory of transformation of motions and the theory of inversion with a description of certain designs of inverters, gives a considerable amount of material, in particular also for the design of the mechanisms indicated. However, this work by no means exhausts the question as a whole. The book by V. von Bool [1] is mainly descriptive in character, presenting the theory of separate mechanisms. Some questions are set forth in reference literature [4], etc.

In such works as the candidate dissertation of G. P. Vyatkin [6], which includes consideration of many designs of mechanisms, in particular for tracing ellipses; the candidate dissertation of O. V. Khotimskaya [7], with a description of a mechanism for tracing curves of the second order; in the works of G. A. Ushakov [9], G. B. Valts [8], and others, there are designs of various drawing mechanisms with a characterization of the operation of some of them; but all these mechanisms are of the plane type and, as was already indicated in the author’s article [5], are distinguished by their complexity and cumbersomeness. Spatial mechanisms for tracing ellipses, although possessing a number of substantial advantages over plane ones, have nevertheless so far been insufficiently investigated.

In our further consideration of mechanisms we shall confine ourselves to mechanisms of the spatial type.

Classification of Spatial Mechanisms

As the basis for the classification of spatial mechanisms we shall take the character of those theoretical errors which are inherent in them because of their structural shortcomings. This considerably simplifies the theoretical investigations of the indicated mechanisms.

All existing spatial mechanisms for tracing ellipses may be divided into two parts. The first part reproduces or creates the curve of an ellipse. This is ensured either by the parallel projection of an inclined circular disk onto the plane of the drawing [10, 19], or by the presence of a special elliptical drum [21] or template [18]. The second part serves to change the size of the ellipse reproduced by the first part.

The author knows of 17 designs of the mechanisms under consideration. All of them may be divided into the following four groups.

The first group includes mechanisms in which the change in the size of the ellipse reproduced by the first part of the mechanism is carried out by the method of similarity. V. von Bool [1] gives a certain characterization of the operation of mechanisms of this group. Describing the design of a three-ended compass, he notes that changing the size of the ellipse with one and the same ratio of axes (one and the same template) is possible only within small limits, since in this case the mechanism does not give the exact form of an ellipse. The mechanisms of this group were also partially investigated by the author in his article [5], where it was analytically established that such mechanisms indeed do not ensure the formation of a geometrically exact ellipse curve. The mechanisms of this group include: 1. Robert Keilert’s elliptical compass [18]. 2. B. D. Pets’s device for tracing curves [16]. 3. The three-ended compass [1].

The second group includes mechanisms whose action is based on changing the size of the ellipse reproduced...

by the first part of the mechanism, by laying off a constant straight-line segment along the normal to this ellipse. Such mechanisms include: 1) the device for drawing curvilinear figures concentric with a given figure, by A. I. Shershnev [17]; 2) the elliptic compass of Cornelius Ott [19], presented in two variants.

The third group includes mechanisms that change the size of the ellipse reproduced by the first part of the mechanisms by laying off a constant straight-line segment along the radius vector to this ellipse. Such mechanisms include: 1) the compass for drawing ellipses and other curves by I. Günther [20]; 2) the ellipsograph of P. P. Nikolaenko [21]; 3) the compass for drawing ellipses by N. A. Kozhemyakin [22]; 4) the device for drawing ellipses by V. F. Gumilevsky [23].

The fourth group includes all the remaining mechanisms. Unlike the mechanisms of the first three groups, these mechanisms are free of theoretical errors. Among the mechanisms of this group there are those that are common in their principle of operation. Therefore, they in turn may be divided into two subgroups. The first subgroup includes mechanisms whose action is based on the rolling of one circle inside another fixed circle, with the diameter of the moving circle being half that of the fixed one. It is known that in this case all points both inside the moving circle and rigidly connected with it outside its limits describe ellipses [1]. Such mechanisms include: 1) Brown’s ellipsograph [1]; 2) the ellipsograph of M. I. Andreev [11]; 3) an elliptic compass with a parallel crank mechanism and two equal meshing gears [12]. The second subgroup includes mechanisms whose action is based on the most varied principles: 1) Burgard’s elliptic compass [1]; 2) the device for drawing ellipses by S. E. Smotrov [13]; 3) the compass for drawing ellipses by S. I. Gadalin [14]; 4) the device for drawing ellipses and other curves by V. I. Domashev [15]; 5) the author’s elliptic compass, presented in two variants [10].

Analysis of the Theoretical Accuracy of the Operation of Mechanisms of the Second Group

As was already indicated [5], the overwhelming majority of mechanisms for drawing ellipses do not ensure the drawing of exact elliptic curves because of their design shortcomings. However, it would be wrong to declare them completely unsuitable. From this point of view, investigating the character of the curves formed by these mechanisms and comparing them with geometrically exact curves will determine the areas of their practical application and will provide initial data for the transition to more advanced designs. In the present article, a study is made of the curves obtained through the use of mechanisms of the second group. In article [5], no specific determination was made of the dependence that exists between the parameters and dimensions of the curve being drawn and the maximum error of the mechanisms. To solve the indicated problem in general form, let us introduce the following notation (Fig. 1):

Fig. 1.

\[ OD = a, \qquad OF = b, \]

where \(a\) and \(b\) are the semiaxes of ellipse 1, obtained by the first part of the mechanisms. The line \(ABA_n\) is the normal to ellipse 1. Curve 2 is an ellipse with semiaxes \(a+d\) and \(b+d\). Curve 3 is the equidistant curve of ellipse 1, obtained by means of the mechanisms under consideration by laying off a constant segment of length \(d\) along the normals to ellipse 1, i.e.,

\[ DE=AA_n=GF=d. \]

Let us write the equations of ellipses 1 and 2:

\[ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1,\quad \frac{X^2}{(a+d)^2}+\frac{Y^2}{(b+d)^2}=1. \tag{1} \]

The distance between any two points, one of which lies on ellipse 1 and the other on ellipse 2, is determined by the equation

\[ L=\sqrt{(X-x)^2+(Y-y)^2}. \tag{2} \]

If the principle of operation of the mechanisms is taken into account, then their theoretical error should be determined along the normal to ellipse 1. However, the introduction of an additional equation of the normal considerably complicates the solution of the problem posed. Therefore, with a negligibly small error not exceeding \(0.01\%\), the error of the mechanisms \(p\) may be determined from the following equation:

\[ A_nB=p=d-L_{\min}, \tag{3} \]

where \(L_{\min}\) is the least distance between two current points \(A\) and \(B\), of which point \(A\) lies on ellipse 1 and point \(B\) on ellipse 2. In this case the error of the mechanisms \(p\) will have its maximum value, which is of greatest interest to us. To determine \(L_{\min}\), we apply Lagrange’s method and write the equation of a function of several variables, which will include equations (1, 2) and the undetermined multipliers \(\lambda_1,\lambda_2\) [25]:

\[ \Phi=(X-x)^2+(Y-y)^2+\lambda_1\left[ \frac{X^2}{(a+d)^2}+\frac{Y^2}{(b+d)^2}-1 \right] +\lambda_2\left[ \frac{x^2}{a^2}+\frac{y^2}{b^2}-1 \right]. \]

Let us take the partial derivatives of this function:

\[ \Phi'_x=-(X-x)+\lambda_2\frac{x}{a^2}=0. \tag{4} \]

\[ \Phi'_y=-(Y-y)+\lambda_2\frac{y}{b^2}=0. \tag{5} \]

\[ \Phi'_X=X-x+\lambda_1\frac{X}{(a+d)^2}=0. \tag{6} \]

\[ \Phi'_Y=Y-y+\lambda_1\frac{Y}{(b+d)^2}=0. \tag{7} \]

From equations (4) and (5) we have

\[ X=\frac{(\lambda_2+a^2)x}{a^2}, \qquad Y=\frac{(\lambda_2+b^2)y}{b^2}. \tag{8} \]

Substituting these values of \(X\) and \(Y\) into equations (6) and (7), we obtain

\[ \lambda_1=-\frac{\lambda_2(a+d)^2}{\lambda_2+a^2}, \qquad \lambda_1=-\frac{\lambda_2(b+d)^2}{\lambda_2+b^2}. \]

From these equations we obtain the value of one of the undetermined multipliers

\[ \lambda_2=\frac{a^2(b+d)^2-b^2(a+d)^2}{(a+d)^2-(b+d)^2}. \tag{9} \]

To determine the value of \(x\) for \(L_{\min}\), in equation (1) we substitute the value of \(y\) from (8) and obtain

\[ \frac{x^2}{a^2}+\frac{Y^2b^2}{(\lambda_2+b^2)^2}=1. \]

Instead of \(Y\), we substitute its value from (1)

\[ \frac{x^2}{a^2} + \frac{\left[(a+d)^2(b+d)^2-X^2(b+d)^2\right]b^2} {(a+d)^2(\lambda_2+b^2)^2} =1. \]

Substituting the value of \(X\) from (8) into this equation, we obtain

\[ x^2= \frac{ a^4\left[(a+d)^2(\lambda_2+b^2)^2-b^2(b+d)^2\right] }{ a^2(a+d)^2(\lambda_2+b^2)^2-b^2(b+d)^2(\lambda_2+a^2) }. \tag{10} \]

To obtain the value of \(L_{\min}\), we multiply equation (4) by \((X-x)\), and equation (5) by \((Y-y)\). Adding the obtained results, we get

\[ (X-x)^2+(Y-y)^2 = \lambda_2 \left[ -\frac{x}{a^2}(X-x)+\frac{y}{b^2}(Y-y) \right]. \]

The left-hand side of this equation, according to (2), is the value of \(L_{\min}^2\). Substituting into its right-hand side the values of \(X\) and \(Y\) from (8), we obtain

\[ L_{\min}^2 = \lambda_2 \left[ -\frac{(\lambda_2+a^2)x^2}{a^4} + \frac{(\lambda_2+b^2)y^2}{b^4} - \left( \frac{x^2}{a^2}+\frac{y^2}{b^2} \right) \right]. \]

Noting that the value of

\[ \left(\frac{x^2}{a^2}+\frac{y^2}{b^2}\right) \]

on the basis of (1) is equal to unity, and substituting the value of \(y^2\) from (1), we obtain the expression

\[ L_{\min}=\frac{\lambda_2}{a^2b}\sqrt{a^4-x^2(a^2-b^2)}. \tag{11} \]

Thus, the error of the mechanisms \(p\) is determined by applying equations (9), (10), (11) and, as the final one, (3).

To simplify the solution of the problem as applied to one or another specific example, it is necessary, after specifying the parameters \(a\), \(b\), and \(d\), to calculate in advance the quantities from the above-mentioned formulas, tabulate them, and express them graphically. We have carried out the calculation of the desired quantities for axis ratios

\[ \frac{a}{b}=\frac{2}{1}, \qquad \frac{a}{b}=\frac{3}{1}, \qquad \frac{a}{b}=\frac{4}{1} \]

and

\[ \frac{a}{b}=\frac{5}{1} \]

with \(d\) varying from \(0\) to \(10\). For the axis ratio \(\frac{a}{b}=\frac{4}{1}\), a graph has been plotted (Fig. 2). As can be seen from the graph, the absolute error of the mechanisms \(p\) increases more slowly than \(d\), and the curve asymptotically approaches a certain limiting value \(p_{\max}\), which is easily and sufficiently accurately determined by the known method of finding the steady-state temperature of an engine [27]. To the left of the graph, this method has been used to determine \(p_{\max}\), and the essence of this method is also shown there.

Fig. 2.

For determining the absolute error of the mechanisms at intermediate values of \(a\) and \(d\), a summary graph is given in Fig. 3. In this case, the absolute error should be determined by applying the method of interpolation of the distance between the adjacent curves and at those values of \(d\) which will represent the curves between themselves (Fig. 3) of interest. A comparison of the results obtained indicates that, with an increase in the elongation of the ellipse, the maximum absolute error of the mechanisms \(p_{\max}\) increases.

Fig. 3.

In the practice of using parts having an elliptical shape (elliptical cams, cylinders, etc.), what is more often of interest is not the absolute magnitude of the deviation of the form, but the relative one. For this purpose, Fig. 4 presents curves that express such a dependence, where the measure of distortion of the geometric form of the ellipse is taken to be the ratio of the absolute error of the mechanisms \(p\) to the length of the major semiaxis \(a + d\) of the curve being drawn. For convenience in reading, the relative error of the mechanisms is represented on the graph by a quantity one thousand times greater than its actual value.

As the character of these curves shows, the relative error of the mechanisms, as \(d\) increases, at first grows sharply, and then, after reaching a certain maximum, gradually begins to fall. Curve \(0\, I\, II\, III\, IV\) represents the maximum values of the relative errors of the mechanisms for various ratios of the axes of the curve being drawn. As the elongation of the curve being drawn increases, the value of the relative error increases. With the aid of the curves obtained in Fig. 4, it is possible to determine the relative error of the mechanisms not only for the selected variants of the ratios of the axes of the curve, but also for intermediate values. For this it is again necessary to apply the interpolation method. The author’s attempt to reduce the dependence between \(a\), \(d\), and \(p\), for \(b = \mathrm{const}\), to a form convenient for nomography from aligned points [28], making it possible to determine intermediate values quickly and with sufficient accuracy, did not lead to positive results.

Fig. 4.

For the rapid calculation of intermediate values of the maximum relative errors of the mechanisms \(p_{\mathrm{rel.}\,(\max)}\), as a function of the change in the major semiaxis of the curve being drawn at \(b = \mathrm{const}\), a graph is given in Fig. 5.

The relative error of the mechanisms makes it possible to determine the suitability of the operating principle of the mechanisms of this group for the mechanical machining of elliptical parts by the copying method. Copying elliptical cams and templates according to the second accuracy class ...

according to the accepted tolerances [26], for the ratio of the axes of the ellipse \( \frac{a}{b} \leq 2 \), can be carried out only when \( \frac{d}{b} \leq 0.1 \) (Fig. 4), and according to the third class—when \( \frac{d}{b} \leq 0.15 \). This means that, in practice, mechanical machining of elliptical cams and templates without prior allowance for the relative error of the mechanisms cannot be performed. The same may be said with respect to the mechanical machining of elliptical cylinders for calculating devices, which are machined according to the second accuracy class [3].

Fig. 5.

Example

Given: \(a = 150\,\mathrm{mm}\), \(b = 50\,\mathrm{mm}\), and \(d = 100\,\mathrm{mm}\). Find the absolute and relative deviation of the curve obtained by the mechanisms of the second group from the geometrical form of the ellipse.

Solution. In accordance with Fig. 3, for \(a = 3\), \(b = 1\), and \(d = 2\), we find \(p = 0.130\). Or, for our example, \(p = 0.130 \cdot 50\,\mathrm{mm} = 6.5\,\mathrm{mm}\). Measuring the absolute theoretical error directly on the drawing (Fig. 6), presented at a scale of 1:2, we obtain \(3.35 \times 2 = 6.7\,\mathrm{mm}\). This means that the difference between the analytically calculated and the graphically constructed theoretical error of the instruments is \(0.2\,\mathrm{mm}\) and lies within the limits of accuracy of graphical constructions.

The relative error, in accordance with Fig. 4, will be \(p_{\mathrm{rel}} = 0.026\). According to the accepted tolerances [26, 3], mechanical machining of parts of elliptical form according to the operating principle of the mechanisms of the second group, by the second accuracy class, cannot be carried out for our example.

As a result of the analysis of the theoretical accuracy of operation of the mechanisms of the second group, the following should be noted.

1. With the aid of mathematical analysis, the problem of determining theoretical errors has been solved in general form.

2. Graphical investigations, which facilitate the mathematical calculations, make it possible to determine the theoretical errors of mechanisms quickly and with sufficient accuracy.

3. The grapho-analytical investigations carried out will help determine the advantages and shortcomings of the mechanisms of the second group and find that technical field in which, according to the accepted tolerances and conditions, they may be applied.

REFERENCES

1. V. von Bool. Instruments and Devices for Drawing, with an Exposition of Their Theory, publ. Moscow Department of the Imperial Russian Technical Society, Moscow, 1892.

2. N. E. Delone. Transmission of Rotation and Mechanical Drawing of Curves by Hinged-Lever Mechanisms, Kirshbaum Printing House, Palace Square, St. Petersburg, 1894.

Fig. 6.

Visible labels in the figure:

• \(x\)
• \(y\)
• \(O\)
• \(C\)
• \(D\)
• \(E\)
• \(F\)
• \(A(x,y)\)
• \(B(x,y)\)
• \(\Delta a_n(x,y)\)
• ellipse
• \(1\)
• \(2\)
• \(3\)

Parameters shown:

\[ a = OD = 150\ \text{mm}, \]

\[ b = OF = 50\ \text{mm}, \]

\[ d = DE = 400 = FG = 100\ \text{mm}, \]

\[ \varphi = 80^\circ,\quad a_n = 67\ \text{mm}. \]

1. S. O. Dobrogurskii, V. K. Titov, Counting-and-solving devices, Oborongiz, Moscow, 1953.

2. H. Dubbel, Taschenbuch für den Maschinenbau, I, Berlin, 1949.

3. E. I. Toropygin, “Investigation of elliptical compasses,” Proceedings of the Tomsk Polytechnic Institute, vol. 85, Tomsk, 1957.

4. L. Sh. Vaitkun, “A mechanism for drawing plane curves and axonometric images,” Candidate dissertation, Moscow, 1949.

5. O. V. Khotimskaia, “Investigation of the application of second-order curves in certain technical problems,” Candidate dissertation, Moscow, 1951.

6. G. V. Val’ts, “Some questions of the mechanization of the drafting process,” in the collection Methods of Descriptive Geometry and Its Applications, Moscow, Gostekhizdat, 1955.

7. G. A. Ushakov, “Mechanisms for the visual representation of mine workings in affine and axonometric projections,” in the collection Methods of Descriptive Geometry and Its Applications, Moscow, Gostekhizdat, 1955.

8. E. I. Toropygin, “Design of elliptical compasses,” Proceedings of the Tomsk Polytechnic Institute, Institute Press, vol. 85, Tomsk, 1957.

9. M. I. Andreev, “An ellipsograph of new design,” Shipbuilding, No. 2, Sudpromgiz, 1951.

10. V. Meier zur Kapellen, Mathematical Instruments, Foreign Literature Publishing House, Moscow, 1950.

11. S. E. Smotrov, “Device for drawing ellipses,” Soviet patent No. 19793, class 42 a, 12, of January 31, 1929.

12. S. I. Gadalin and T. I. Gadalina, “Compass for drawing ellipses,” Soviet patent No. 20324, class 42 a, 12, of April 4, 1929.

13. V. I. Domashev, “Device for drawing ellipses and other curves,” Soviet patent No. 16003, class 42 a, of July 13, 1929.

14. B. D. Pets, “Device for drawing various curves,” Soviet author’s certificate No. 35377, class 42 a, 14, of September 20, 1932.

15. A. P. Lebedev, “Device for drawing curvilinear figures concentric with a given figure,” Soviet patent No. 2871, class 42 a, of December 4, 1922.

16. Robert Kahlert in Lauf near Nürnberg, “Zirkel zum Zeichnen von Ellipsen und ähnlichen Kurven.” Patented in the German Reich No. 483720 of March 7, 1928.

17. Cornelius Ott in Stuttgart-Hofen, “Gerät zum Zeichnen Ellipsen.” Patented in the German Reich No. 603712 of June 8, 1933.

18. I. Gonter, “Compass for drawing ellipses and other curves,” Soviet patent No. 11464, class 42 a, 12, of September 30, 1929.

19. P. P. Nikolaenko, “Ellipsograph,” Soviet patent No. 9695, class 42 a, of March 29, 1928.

20. N. A. Kozhemiakin, “Compass for drawing ellipses,” Soviet author’s certificate No. 42693, class 42 a, 12, of September 3, 1934.

21. V. F. Gumnilevskii, “Device for drawing ellipses,” Soviet author’s certificate No. 28022, class 42 a, 12, of October 31, 1932.

22. S. B. Galustian and N. M. Sheidaev, Second-order curves and their application in engineering and natural science, AKII Press, Baku, 1936.

23. G. M. Fichtenholz, Fundamentals of Mathematical Analysis, vol. 2, Gostekhizdat, Moscow, 1956.

24. S. V. Grumm-Grzhimailo, Selection of tolerances in mechanical engineering, Mashgiz, Moscow, 1950.

25. NKTP Glavenergo. TsES. Electrical engineering rules and standards, ONTI, Gosenergoizdat, Moscow–Leningrad, 1933.

26. N. M. Gersevanov, Theory and Construction of Engineering Nomograms, ONTI—NKTP—USSR, Moscow–Leningrad, 1937.