CONSTRUCTION OF ELLIPTIC COMPASSES

E. I. TOROPYGIN

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

Graphic work performed at aircraft, automobile, and shipbuilding plants is often accompanied by the drawing of elliptic curves. Axonometric drawings at machine-building plants, for the most part, also require the drawing of elliptic curves. Naturally, elliptic compasses acquire considerable importance in this case.

Most existing instruments of this type \[1, 2, 3\] have serious shortcomings and in a number of cases do not fulfill their direct purpose \[5\].

The aim of the present work is to acquaint the reader with constructions of elliptic compasses developed by the author.

First construction

The operation of the instrument (Fig. 1) is based on the parallel projection of inclined circular disks 6 onto the plane of the drawing. The angle of inclination of the disks is adjusted by means of rods 7 and a sleeve 8, sliding along the fixed leg of the compass 4 and secured by screw 9.

The vertical rod 5, rigidly connected with two sliding bushings 1, performs parallel motions, since it moves along parallel guides 3. At the same time, rod 5 is continuously pressed against the periphery of the inclined disks by springs 2 and, in a section perpendicular to leg 4, describes an elliptic curve. The ratio of the axes of the ellipse is adjusted by the angle of inclination of the disks.

The guides 3, rigidly connected to one another by frame 10, together with rod 5 move around the fixed disks 6.

Point \(e\), located on the continuation of the axis of rod 5, like rod 5 itself, describes an ellipse. This point terminates the first part of the instrument, intended for reproducing an ellipse.

The second part of the instrument is a hinged parallelogram \(abcd\), serving to change the size of the ellipse reproduced by the first part.

Let us consider the operation of this hinged parallelogram. For this purpose we shall take it separately (Fig. 2).

The points \(o\), \(q\), and \(e\) are situated on one straight line; point \(o\) is the fixed pole, point \(e\) is the point describing the ellipse, and point \(q\) is the writing element.

When point \(e\) is displaced to point \(e_1\), point \(o\) does not change its position; the distances \(ob\), \(bc\), \(cd\), \(be\), \(ce\), \(cq\), and \(qd\) remain constant, and therefore, to obtain point \(b_1\), from point \(o\) with radius \(ob\) we draw one arc, and from point \(e_1\) with radius \(be\) we draw a second arc. The intersection of these arcs

will give point \(b_1\)—the new position of hinge \(b\). Laying off along the line \(b_1 e_1\) the length \(bc\), we obtain point \(c_1\)—the new position of hinge \(c\). Drawing from points \(c_1\) and \(o\) parallels to the straight lines \(ob_1\) and \(b_1 e_1\), we obtain the new position of hinge \(d\)—point \(d_1\). To determine the new position of point \(q\), on the line \(c_1 d_1\) lay off from point \(c_1\) the length \(cq\).

Owing to the parallelism of the arms \(ob\) and \(dc\), the triangles \(obe\) and \(qce\) are similar, and therefore:

\[ \frac{ob}{qc}=\frac{be}{ce}=\frac{oe}{qe}. \tag{1} \]

Fig. 1

Fig. 2

Making the corresponding substitution, we obtain:

\[ \frac{ob_1}{q_1c_1}=\frac{b_1e_1}{c_1e_1}=\frac{oe}{qe}. \tag{2} \]

Taking into account that, during the operation of the parallelogram, the condition of parallelism of its opposite sides is always satisfied, we obtain: \(\Delta ob_1e_1 \sim \Delta q_1c_1e_1\). Hence the sides \(oe_1\) and \(q_1e_1\) form one straight line.

On this basis, and from (2), one may write the proportion

\[ \frac{oe}{qe}=\frac{oe_1}{q_1e_1}. \tag{3} \]

Further, from Fig. 2 it is seen that the triangles \(oee_1\) and \(oqq_1\) have a common angle enclosed between proportional sides, and therefore they are similar. From their similarity it follows that

\[ ee_1 \parallel qq_1 \quad \text{and} \quad \frac{ee_1}{qq_1}=\frac{oe}{oq}. \]

For an infinitely small magnitude of the line \(ee_1\), it coincides with the curve onto which some portion of the inclined disks is projected, while the line \(qq_1\) coincides with the curve obtained by a subsequent change in the magnitude of the first curve. Consequently, the second part of the instrument traces ellipses similar to those of the first part and with the same ratio of axes as that given by the inclined disks, i.e., the instrument works without distorting the curve.

The parallelogram used in Figs. 1 and 2 corresponds to a reduction of the ellipse reproduced by the first part of the instrument. If the points \(e\) and \(q\) are interchanged, the instrument will produce an enlargement of the ellipse.

Use of the instrument consists in first setting the inclined disks in the position corresponding to the required ratio of axes. For this purpose the coupling 8 (Fig. 1) is set opposite the corresponding division marked on the fixed leg 4 and is secured by screw 9. Then the sliders at the points \(c\), \(q\), and \(d\) are set opposite the divisions marked on the levers \(be\), \(cd\), and \(or\), corresponding to the required size of the ellipse, and are secured with screws.

Second construction

The second construction (Fig. 3) differs from the first in the design of the second part of the instrument, in the direction of its simplification. Here the system of hinged levers of the second part is made with obligatory observance of the following conditions:

1) \(ce = oc\),

2) \(od = dq\),

3) the points \(c, o, d\) and \(q, o, e\), respectively, lie on one straight line.

The point \(e\), relative to the fixed part of the instrument (the central leg with the inclined disks and the two lateral legs 4), describes an elliptical curve, while relative to the movable part of the instrument it moves rectilinearly, parallel to the guides 2. The lever \(dc\), receiving motion from the point \(e\), rotates about the point \(o\) and transmits rotation to the hinge \(d\) (Fig. 4). The hinge \(d\) transmits motion to the slider \(q\), which carries the writing element. The point \(q\) moves along an elliptical curve. The latter statement is proved as follows.

In connection with the conditions (1–3) adopted above, the triangles \(oce\) and \(odq\) prove to be similar. Consequently:

\[ \frac{oc}{od} = \frac{ce}{dq} = \frac{oe}{oq}. \tag{4} \]

When the point \(e\) is displaced to the point \(e_1\), the triangles \(oc_1e_1\) and \(od_1q_1\) will likewise be similar, and therefore:

\[ \frac{oc_1}{od_1} = \frac{c_1e_1}{d_1q_1} = \frac{oe_1}{oq_1}. \tag{5} \]

Since the lengths \(oc\), \(od\), \(ce\), \(dq\) remain constant during operation of the instrument, replacing \(oc_1\) by \(oc\) and \(od_1\) by \(od\), from (4) and (5) we obtain:

\[ \frac{oe}{oq} = \frac{oe_1}{oq_1}. \]

Thus point \(q\) describes an ellipse similar to the ellipse described by point \(e\), that is, with the same ratio of axes and without distortion. The change in the size of the ellipse will be determined by the ratio of the arms \(\frac{od}{oc}\).

The simplification of construction 2 in comparison with construction 1 consists in the fact that in this construction the size of the ellipse is set at one point \(d\) instead of the 3 points of the first construction. The arms \(od\) and \(dq\) must be provided with the corresponding graduations indicating the size of the ellipses.

From experience in operating the first construction, the necessity was revealed of installing two additional side legs to ensure the stability of the instrument.

Thus, the constructions of elliptic compasses proposed by the author ensure the production of an ellipse without distortion.

Fig. 3

Fig. 4

The developed constructions may find application for drawing ellipses in axonometric projection, where it is often necessary to draw a large number of ellipses of different sizes with one and the same ratio of axes. In this case coupling 9 (Figs. 1 and 3) remains in place, and the entire adjustment of the instruments is carried out either by three screws of the articulated parallelogram in the first construction, or by one screw of the system of articulated arms in the second construction. In addition, the instruments may be used in any case where it is necessary to obtain an accurate elliptical curve.

References

1. Robert Kahlert in Lauf near Nürnberg. Compass for drawing ellipses and similar curves. Patented in the German Reich, No. 483720, March 7, 1928.

2. P. P. Nikolaenko. Ellipsograph. Soviet patent No. 9695, class 42A, 12, dated March 29, 1928.

3. Cornelius Ott in Stuttgart—Hofen. Device for drawing ellipses. Patented in the German Reich, No. 603712, June 8, 1933.

4. V. Meyer zur Capellen. Mathematical Instruments. Moscow: Foreign Literature Publishing House, 1950.

5. E. I. Toropygin. “Investigation of Elliptic Compasses.” Article included in the present collection, p. 366.