STUDY OF ELLIPTIC COMPASSES

E. I. TOROPYGIN

(Presented by Prof. P. M. Alabuzhev, Doctor of Technical Sciences)

Instruments for drawing ellipses may be divided into two principal types:

1. Planar instruments.
2. Compass-type instruments (elliptic compasses).

The first instruments are distinguished by their bulkiness and comparative complexity, but from them it is easier to proceed to more complex instruments that make it possible to construct, for example, axonometry, perspective, shadows, etc.

Elliptic compasses are distinguished by their small overall dimensions, compactness, and ease of use.

The present work has been written in order to reveal errors relating to existing elliptic compasses and to establish the conditions that must be observed in designing them.

Let us consider three existing designs.

Fig. 1

The first design is by the German author Robert Keilert, known from German patent No. 483720, class 42A, 12, of March 7, 1928.

The diagram of the instrument is given in Fig. 1. The operating principle of the instrument amounts to the following: rod 6, which carries the writing element, rolls around replaceable elliptic template disks 1. In this process rod 6 is constantly pressed against template disks 1 by means of spring 4 and, being hinged to lever 7, which is provided with a handle for rotating the rotary system, turns about the hinge point \(p\).

The required ratio of the axes of the ellipse is obtained by replaceable template disks. By moving the disks along the fixed leg 3 and subsequently fastening them on this same leg by means of screw 2, the author intended to obtain a change in the size of the curve. In reality, however, the instrument cannot enlarge or reduce the curve reproduced by the template disks without distorting it, for the following reason.

The conditional center of rotation \(o\) of rod \(pq\) (such will be the extended position of rod \(pq\) to its intersection with the fixed leg of compass 3), during operation continuously moves along leg \(ts\) (Fig. 2).

If \(et\) corresponds to the minor axis of the given ellipse, then its enlargement is determined by the ratio \(\dfrac{oq}{oe}\), since \(et \parallel qs\) and \(\Delta oqs \sim \Delta oet\). From the similarity of the triangles it follows that

\[ \frac{qs}{et}=\frac{oq}{oe}=\frac{os}{ot}. \tag{1} \]

Neglecting the small vertical displacement of the point \(p\) and taking into account that the point \(p\) lies on the rigid lever 7, the enlargement of the major axis, if the quantity \(e_1t\) is taken for it, is determined by the ratio \(\dfrac{o_1q_1}{o_1e_1}\), since

\[ \frac{q_1s}{e_1t}=\frac{o_1q_1}{o_1e_1}=\frac{o_1s}{o_1t}. \tag{2} \]

Comparing equations (1) and (2), we obtain

\[ \frac{oq}{oe}\ne\frac{o_1q_1}{o_1e_1}. \]

Consequently, the curve is obtained with distortion.
To compensate for the indicated shortcoming, the point \(p\) of rotation of the rod \(pq\) should be brought into coincidence with the point \(o\) and fixed so that it does not move vertically, while the rod \(pq\) should be provided with a device allowing the length \(pq\) to be varied (Fig. 3).

Fig. 2

Fig. 3

Suppose that \(et\) and \(e_1t\) are respectively equal to the minor and major axes of the ellipse. Then the enlargement of the minor axis is determined by the ratio:

\[ \frac{qs}{et}=\frac{oq}{oe}=\frac{os}{ot}, \tag{3} \]

and of the major axis, respectively:

\[ \frac{q_1s}{e_1t}=\frac{oq_1}{oe_1}=\frac{os}{ot}. \tag{4} \]

Hence

\[ \frac{q s}{e t}=\frac{q_1 s}{e_1 t} \quad \text{or} \quad \frac{o q}{o e}=\frac{o q_1}{o e_1}. \]

Thus in this case no distortion is obtained.

The second design is by the Soviet author P. P. Nikolaenko—Soviet patent No. 9695, class 42A, 12, of March 29, 1928 (Fig. 4).

The principle of operation of the design is as follows: rod 7, carrying the writing element 11, rotating about the fixed elliptical drum 4, remains parallel to the fixed leg 1. This is achieved by the presence of two parallel springs 6, attached at one end to rod 7 and at the other to sleeve 5, freely seated on leg 1. Springs 6 continuously press rod 7, through screw 9, against the elliptical drum 4. In its lower section the elliptical drum is a circle, and in its upper section an ellipse; moreover, the major axis of the ellipse, throughout the entire length of the drum, is a constant quantity and is equal to the diameter of the circle of the lower section, while the minor axis, starting from the lower section, continuously decreases and has its minimum value at the upper base of the drum.

The instrument gives an exact ellipse curve only in the case when the point of contact of screw 9 with drum 4 (point \(e\)) and the writing point (point \(q\)) are on one vertical line, that is, if the writing point \(q\) traces an ellipse reproduced in the section of the drum at the level of point \(e\) without changing its size.

Fig. 4

Changing its size, however, carried out with the aid of screw 9, leads to a distortion of the character of the curve, since the quantity by which the radius vectors of the ellipse being drawn are changed does not change during the process of drawing it.

Besides distorting the curve, the instrument has a second drawback, which consists in the fact that when the size of the curve is changed, a change in the ratio of its axes is obtained, that is,

\[ \frac{a}{b}\ne \frac{a+b_1}{b+b_1}, \]

where \(b_1\) is a constant quantity by which the parameters \(a\) and \(b\) of the ellipse reproduced by the elliptical drum are changed. From this point of view, the use of the instrument is very limited.

The instrument of P. P. Nikolaenko can be used only in those cases where great accuracy is not required and where the size of the curves does not differ very much from the curves reproduced by the elliptical drum.

The third design is by the German author Cornelius Otto, known from German patent No. 603712, class 42a, 12, of June 8, 1933 (Fig. 5).

The principle of operation of the instrument reduces to the fact that the curve of the ellipse

is reproduced by projecting the rotating inclined circular disk 1 onto the drawing plane, and its magnitude is varied by screw 5. Rod 4 performs a parallel motion, since it rests on two bushings freely seated on two parallel guides 7 and, during its motion, is constantly pressed against the perimeter of the inclined disk 1 by means of spring 3.

Unlike the instrument of P. P. Nikolaenko, here, when the magnitude of the ellipse reproduced by the inclined circular disk 1 is changed, the constant magnitude \(b_1\) is laid off not along the radius vector, but along the normal to the curve (Fig. 6). This is accomplished by means of two additional rods 9 (besides the main rod 4), connected with the main rod by two flat springs 8. Rods 9, like the main rod 4, are constantly pressed against the perimeter of the inclined disk. Since springs 8 are identical to one another in size and shape, they ensure a constant perpendicular position of the axis of the screw to the tangent drawn to the perimeter of the inclined disk at the point where it touches the main rod.

Consider the question of the range of application of this instrument. Denote the angle of inclination of the inclined disk to the drawing plane by

Fig. 5

Fig. 6

\(\alpha\) (Fig. 7), the semiaxes of the ellipse into which the disk is projected by \(a\) (the larger) and \(b\) (the smaller); then the semiaxes of the curve being drawn will be respectively equal to: \((a+b_1)\) and \((b+b_1)\).

Suppose that it is required to enlarge the ellipse into which the inclined disk is projected, with the ratio of axes

\[ \frac{2a}{2b}=2 \]

by a factor of two, that is, so that the following relations hold:

\[ a+b_1=2a \tag{5} \]

and

\[ \frac{a+b_1}{b+b_1}=2. \tag{6} \]

The angle \(\alpha\) will have to be changed. With the change of the angle \(\alpha\), the magnitude \(b\) will also change. Let us find their new values.

From equation 5 we find that \(b_1=a\).

We obtain the quantity \(b_1\) by means of screw 5. But since screw 5 is rigid, the same quantity \(b_1=a\) will also be laid off along the minor axis of the ellipse being drawn.

Substituting the value \(b_1=a\) into equation (6), we obtain: \(b=0\).

To obtain such a value of \(b\), disk 1 must be placed in a vertical position, which is practically impossible to achieve.

Consequently, the range of application of this instrument, as also of the P. P. Nikolaenko instrument, is very limited.

Let us analyze the character of the curve produced by the instrument. For this purpose we introduce the following notation (Fig. 8):

\[ ON=a;\qquad OM=b;\qquad NC=MF=b_1. \]

Fig. 7                Fig. 8

Let us take curve 1 to be the ellipse into which the inclined disk is projected.

Curve 3 is also an ellipse, with semiaxes \((a+b_1)\) and \((b+b_1)\).

Curve 4 is the parallel, or equidistant [1], to ellipse 1, that is, the curve obtained by means of the instrument.

The equation of ellipse 1 for an arbitrary point \(A\) in parametric form is written as follows:

\[ \begin{aligned} x&=a\cos\varphi,\\ y&=b\sin\varphi \end{aligned} \tag{7} \]

The relation between the angles \(\psi\) and \(\varphi\) is determined as follows:

\[ \tg\psi=\frac{AG}{OG}=\frac{y}{x}. \]

Substituting into this equation the values of \(x\) and \(y\) from equation (7), we obtain:

\[ \tg\psi=\frac{b}{a}\tg\varphi . \tag{8} \]

Through point \(A\) of ellipse 1 we draw the normal to the ellipse. If on this normal, from point \(A\), we lay off the distance \(b_1\), then we obtain the point \(A^*(x^*,y^*)\) on the parallel curve 4. This same normal also intersects curve 3, that is,

ellipse, constructed in the same axes as the parallel curve, also gives the point \(A'(x',y')\).

To express the angle of inclination of the normal to the axis \(Ox\) (the angle \(\gamma\)) in terms of the angle \(\psi\), let us write the equation of the normal:

\[ a^{2}y(x'-x)=b^{2}x(y'-y). \]

Transform this equation:

\[ 1=\frac{b^{2}}{a^{2}}\cdot\frac{x}{y}\cdot\frac{y'-y}{x'-x}. \tag{9} \]

But

\[ \frac{x}{y}=\frac{1}{\operatorname{tg}\psi},\qquad \text{and}\qquad \frac{y'-y}{x'-x}=\operatorname{tg}\gamma \quad (\text{Fig. 8}). \]

Substituting these values into equation (9), we obtain:

\[ \operatorname{tg}\gamma=\frac{a^{2}}{b^{2}}\operatorname{tg}\psi . \tag{10} \]

Now let us write the equation of the parallel curve:

\[ \begin{aligned} x^{*}&=x+b_{1}\cos\gamma,\\ y^{*}&=y+b_{1}\sin\gamma . \end{aligned} \tag{11} \]

To determine the error of the instrument, it is necessary to determine the distance \(A'A^{*}\).

We shall bring the equation of the normal to the following form:

\[ y'-y=\operatorname{tg}\gamma\,(x'-x). \tag{12} \]

Then the equation of ellipse 3 will have the form:

\[ \frac{(x')^{2}}{(a+b_{1})^{2}}+\frac{(y')^{2}}{(b+b_{1})^{2}}=1. \tag{13} \]

Solving these two equations (12 and 13) jointly, we find the value of \(x'\):

\[ x'= \frac{\operatorname{tg}\gamma\,(x\operatorname{tg}\gamma-y)(a+b_{1})^{2}} {(b+b_{1})^{2}+(a+b_{1})^{2}\operatorname{tg}^{2}\gamma} \pm \]

\[ \pm \sqrt{ \frac{ \operatorname{tg}^{2}\gamma\,(x\operatorname{tg}\gamma-y)^{2}(a+b_{1})^{4} +(a+b_{1})^{2}\bigl[(b+b_{1})^{2}+(a+b_{1})^{2}\operatorname{tg}^{2}\gamma\bigr]\bigl[(b+b_{1})^{2}-(x\operatorname{tg}\gamma-y)^{2}\bigr] } {(b+b_{1})^{2}+(a+b_{1})^{2}\operatorname{tg}^{2}\gamma} }. \tag{14} \]

From equation (12) we find the value of \(y'\):

\[ y'=y+\operatorname{tg}\gamma\,(x'-x). \tag{15} \]

Let us calculate the error of the instrument for a concrete example:

\[ a=274\ \text{mm},\qquad b=98\ \text{mm},\qquad b_{1}=57\ \text{mm},\qquad \psi=15^\circ. \]

From (8), for the above values of \(a\), \(b\), \(b_{1}\), and \(\psi\), we find: \(\varphi=37^\circ\). Determine the coordinates of the point \(A\) of curve 1 by formula (7):

\[ x=274\cdot\cos37^\circ=218\ \text{mm}, \]

\[ y=98\cdot\sin37^\circ=58.8\ \text{mm}. \]

From formula (10) we find \(\gamma=64^\circ30'\).

Substitute the value of the angle \(\gamma\) into formula (11) and find the coordinates of point \(A^*\) of the parallel curve:

\[ x^* = 242.5\ \text{mm}, \]

\[ y^* = 110.3\ \text{mm}. \]

The coordinates of point \(A'\), calculated by formulas (14) and (15), are obtained as:

\[ x_1' = 240.3\ \text{mm}, \qquad y_1' = 105.7\ \text{mm}. \]

Let us calculate the error of the instrument along the normal (the segment \(A^*A'\)):

\[ (A^*A')^2 = (x^* - x_1')^2 + (y^* - y_1')^2. \]

Hence

\[ A^*A' = 5.1\ \text{mm}. \]

Let us consider the second example: \(a = 150\ \text{mm}\), \(b = 50\ \text{mm}\), \(b_1 = 100\ \text{mm}\), \(\psi = 10^\circ\).
From (8) we find: \(\varphi = 28^\circ\).
Determine the coordinates of point \(A(x,y)\):

\[ x = 132.2\ \text{mm}, \qquad y = 23.4\ \text{mm}. \]

From (10) we find the value of the angle \(\gamma\):

\[ \gamma = 57^\circ 45'. \]

By the angle \(\gamma\) we determine the coordinates of point \(A^*\) of parallel curve 4:

\[ x = x + b_1 \cos \gamma = 185.5\ \text{mm}, \qquad y^* = y + b_1 \sin \gamma = 107.8\ \text{mm}. \]

Let us find the coordinates of point \(A'\) of curve 3 by (14) and (15):

\[ x_1' = 181.8\ \text{mm}, \]

\[ y_1' = y + \tan \gamma (x_1' - x) = 102.1\ \text{mm}. \]

The error of the instrument will be

\[ A^*A' = \sqrt{(x^* - x_1')^2 + (y^* - y_1')^2} = 6.8\ \text{mm}. \]

As the examples show, this instrument, like both preceding ones, gives an unavoidable error in drawing the ellipse curve; moreover, as the magnitude \(b_1\) increases, the error increases.

Conclusions

1. Most existing designs provide a very limited range of enlargement or reduction of ellipses.

2. Most existing instruments are structurally arranged in such a way that, in order to change the size of the ellipses being drawn for one and the same ratio of axes, a double adjustment is required. This creates inconvenience, for example, when making axonometric projections.

3. All instruments give curves that are knowingly distorted. Naturally, such instruments do not meet the requirements imposed on them.

References

1. Zhitomirskii O. K., et al. Problems in Higher Geometry, ONTI, NKTP USSR. Moscow–Leningrad, 1937.

2. Bragis V. Analytic Geometry. Uchpedgiz. Moscow, 1937.

3. Meyer, Zur, Kappelen V. Mathematical Instruments. Moscow. Publishing House of Foreign Literature, 1950.

4. Robert Kahlert in Lauf b. Nürnberg. Zirkel zum Zeichnen von Ellipsen und ähnlichen Kurven. Patentiert im Deutschen Reiche No. 483720 vom 7 März 1928 ab.

5. Nikolaenko P. P. Ellipsograph. Soviet patent No. 9695, class 42a, 12, of March 29, 1928.

6. Cornelius Ott in Stuttgart-Hofen. Gerät zum Zeichnen von Ellipsen; patentiert im Deutschen Reiche No. 603712 vom 8 Juni 1933 ab.