US3524257A

US3524257A - Constructing ellipses

Info

Publication number: US3524257A
Application number: US725342A
Authority: US
Inventors: Jan Jakubowski
Original assignee: Standardgraph Filler und Fiebig GmbH
Current assignee: Standardgraph Filler und Fiebig GmbH
Priority date: 1967-05-05
Filing date: 1968-04-30
Publication date: 1970-08-18
Anticipated expiration: 1987-08-18
Also published as: DE1561388A1

Description

Aug. 18, 1970 JAKUBOWSKI 3,524,257

CONSTRUCTING ELLIPSES Filed April 50, 1968 4 Shoots-Sheet 1 Fig. 1

Aug' 18, '1970 J. JAKuBowsKl 3,524,257

coNsTRUcTNG ELLIPSES Filed April 30, 1968 4 Shoots-Sheet 2 Aug. 18, 1970 J. JAKUBowsKI 3,524,257

CONSTRUCTING ELLIPSES Filed April 30, 196B 4 Shoots-Sheet 3 Q lL Filed April 50, 1968 4 Sheets-Sheet 4.

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z QM i a M La, wf R CW N M M W Unted States Patent O 3,524,257 CONSTRUCTING ELLIPSES Jan Jaknbowski, Stuttgart-Hofen, Germany, assignor to Standardgraph Filler & Fiebig G.m.b.H., Geretsried, Upper Bavaria, Germany Filed Apr. 30, 1968, Ser. No. 725,342 Claims priority, applicattnermany, May 5, 1967,

Int. c1. 1;4315/00 U.S. Cl. 33--1 9 Claims ABSTRACT F THE DISCLOSURE There are in existence a number of devices for constructing ellipses. The simplest form of device is a fixed stencil but this has the disadvantage that only ellipses or parts of ellipses which have already been cut in the stencil can be produced.

A further known apparatus is a trammel which is steplessly adjustable so that an infinitely large number of different ellipses can be produced. However the construction is complicated and therefore so expensive that the market is comparatively limited.

There is also the method using a piece of string anchored at both ends which has the advantage of simplicity and cheapness but with it there is the drawback that extensive trial and error is necessary before an ellipse of the desired dimensions can be constructed.

One object of the present invention is to provide a. method and apparatus for drawing ellipses which is more satisfactory than the above-mentioned methods in apparatus.

A further object of the invention is to provide a steplessly adjustable cheap form of ellipse constructing apparatus with which ellipses can evenly be constructed with a pre-set major axis.

A still further object of the invention is to provide an ellipse constructing apparatus which is in the form of simple draughtsmans stencils and is nevertheless steplessly adjustable.

The present invention consists in a method of constructing an ellipse using two families of circular curves, the curves in each family having arithmetically increasing diameters, a common tangent point, and corresponding to the diameters of the other family of curves, the method comprising the superposing of one family of curves onto the other with the centers of curvature of all circles on a single line, one family of curves being 180 offset from the other, and registering a series of points of intersection between curves corresponding to an ellipse. The tangent points can be virtual.

The present invention also consists in an apparatus for constructing an ellipse comprising two relatively movable parts each carrying a family of circular curves on it, the curves in each family having arithmetically increasing diameters with a common tangent point and the diameters of the curves corresponding to those of the other part, the parts being capable of being placed so that the families of curves are superposed with the centers of curvature of the circular curves on a common axis and p ICC;

the families of curves offset by in relation to each other so that a series of points of intersection between curves corresponding to an ellipse can be registered.

Preferably one of the parts is transparent. The parts can be in the form of sheets, for instance transparent sheet. Both parts can be in the form of transparent sheets of plastic foil. One of the parts can be in the form of a stencil adapted for use with a writing instrument for transferring points of intersection onto an underlay. Such a stencil can have slots corresponding to the circular curves, the slots extending symmetrically to each side of an axis passing through all centers of curvature.

The invention will now be described with reference to the accompanying drawings.

FIG. 1 shows a family of circular curves rwhich have a real tangent point, that is to say a point at which the circular curves touch and through which a common tangent could be drawn. The circular curves in each family increase arithmetically in diameter.

FIG. 2 shows a family of identical circular curves offset through 180 in relation to the curves shown in FIG. 1.

FIG. 3 shows the series of points of intersection produced by superposition of the two families of curves shown in FIGS. l and 2, the lines running through the centers of the curves in each family also being superposed.

FIG. 4 shows the superposition of two circular curve families in accordance with FIGS. 1 and 2 with a shorter main or connecting axis than is the case in FIG. 3.

FIG. 5 shows the superposition of the two curve families for producing ellipses with a larger main axis than is the case with FIG. 4.

FIGS. 6, 7, and 8 show curves required for proving that the geometrical construction in accordance with the invention produces ellipses.

FIG. 9 shows two superposed sheets of which each is provided with a family of curves similar to FIGS. l and 2.

FIG. 10 shows a stencil having circular curves in accordance with FIG. 1.

FIGS. 1 and 2 show two families of circular curves which only differ in that one is offset about 180 in relation to the other. In both families of curves the radii increase arithmetically.

FIG. 3 shows the superposition of two families of circular curves of the type shown in FIGS. l and 2 though there are more circular curves than in the families of FIGS. 1 and 2. The two families of curves are superposed in such a manner that the two axes of symmetry connecting the center points of the circles, shown in FIGS. 1 and 2, are superposed. The common axis of symmetry of the families of curves is denoted by reference numeral 12 in FIG. 3.

If one starts, for example, at the point of intersection of two circles with the same radii 1 and then connects all the diagonally opposite points of intersection 2 until a closed curve results, one obtains an ellipse 3, which is shown in the drawing by means of dotted lines with a semi-minor axis extending from point 1 to point 4 and with a semi-major which extends a distance equal to the spacing between points 4 and 5. The major axis of the ellipse 3 extends between points 5 and 6.

For any randomly chosen major axis (the distance between points 5 and 6) there is formed, in the above manner, a series of ellipses 7 with different minor axes; some of these elipses 7 are shown in broken or dotted lines in FIG. 3.

If the distance between points 5 and 6 is changed by parallel movement of the families of circular curves along the axis of symmetry 12, for example by moving sheets on which the two families of curves are respectively drawn in accordance with FIGS. 1 and 2, the upper sheet at least being transparent, it is possible to obtain further series of ellipses with different major axes. An example of such a series of ellipses with two diderent major axes are shown in FIGS. 4 and 5. In FIG. 4 the major axis of the various ellipses, which are also constructed as is the case in FIG. 3, lies between

points

8 and 9. In FIG. 5 the major axis of the family of ellipses, which can also be constructed as is the case in FIG. 3, lies between points and 11.

The proof that the above-described geometrical loci of points represent true ellipses is as follows: it is rst assumed that there are two families of curves in accordance with the equations in which a=abscissa of the common point of a family of curves r1=initial radius of both families of curves Ar=constant increment or decrement in radius (family parameter).

In accordance with the value of Ar and owing to the crossing of the two families of curves a more or less dense series of points of intersection is obtained (FIG. 6).

In this series of points of intersection it is possible to see the geometrical locus of points which consists of a selected intersection point P of a pair of circular curves with the same radii r1 and of all further points of intersection Q of neighbouring pairs of circular curves in which the radius of a circle belonging to such a pair of curves is increased by the amount Ar and the radius of the other circular curves is decreased by the same amount, Ar.

The equation of this geometrical point locus can be derived by eliminating the parameters r1 and Ar from

Equations

1 and 2.

To do this the relationship between r1, a, and b is first found from Equation 1 or Equation 2 by replacement of the coordinates of the point of intersection P in FIG. 7 with a pair of circular curves having the same radii equal to r1(Ar=0),

Furthermore by putting the coordinates of a point of intersection Q of circular curves in accordance with FIG. 8 into

Equations

1 and 2 and eliminating the ordinates, the family parameter Ar is expressed in accordance with the abscissa X of the intersection point Q as follows tl-T1. AT---a X (4) From (3) and (4) we then obtain a2 .b2 AT- 2a2 X 5) By then putting Ar in accordance with Equation 5 and r1 in accordance with Equation 3 for example into Equation 1, one then obtains the sought-after equation of the geometrical locus of points which after shortening and re-arrangement finally gives the equation of an ellipse FIG. 9 shows an an embodiment of the invention in which reference numeral 50 denotes a planar base or sheet on which a family of circular curves'is drawn. On the sheet 50 there is a further sheet or base 51 which is also provided with the above-mentioned family of circular curves. Reference numeral 1'2 denotes the Whole axis of symmetry. By movement of the two sheets 501 and 51 in relation to each other it is possible to arrange for different maior axes for different families of ellipses. The less the distance between the radii of the bases Sil and 51, the smaller the distance between the points of intersection and the more accurate it is to construct the desired ellipse, the two bases or

sheets

50 and 51 can be transparent or preferably at least the upper sheet 51 is transparent.

The manner of operation Iwith an opaque sheet Sil and a transparent sheet 51 is as follows: The axes of symmetry 12 of the two families of circular curves offset from each other by 180 are brought so that one lies exactly on top of the other. By parallel displacement between the two

sheets

50, 51 along the axis of symmetry 12 the distance between the

points

8 and 9, which form the main vertices of the ellipse, is altered in order to make the major axis of the required length. Then the sheets are fixed together, for example by means of removable sticky tape, at two positions. The ellipses required are then registered, for example by copying onto drawings, plates or the like. They can also be photo-copied or registered point by point and then drawn by means of a standard draughtsmans set of curves.

The method in accordance with the invention for producing and drawing ellipses of any required size represents a substantial saving in time and the apparatus used is cheap. The apparatus can consist simply of two printed sheets, for example as shown in FIG. 9. lIn practice it is found that it is suliicient to purchase the upper transparent sheet S1 since it is possible to produce a cheap photo-copy for the lower sheet required.

FIG. 10 shows a stencil 100 for the construction of ellipses. lIt is rectangular and is provided with a series or family of circular curves represented by slots such as 102, 1013, 104. The tangent point is for most curves virtual that is to say they do not extend as far as it. At the point corresponding to the tangent point there is a hole 101 passing through the stencil. Along the axis 12 of symmetry of the family of curves there are longitudinal slots 90. Preferably the stencil is made of transparent synthetic resin.

For the construction of ellipses one can for example use two stencils 100, with curves similar to the curves marked on sheets shown in FIG. 9. The stencils are laid on top of one another so that the individual points can easily be marked or otherwise transferred onto any desired object. It is also possible to use a stencil together with another transparent or opaque sheet having a family of curves .marked on it and then to transfer the ellipse points onto an object.

I claim:

1. An apparatus for constructing an ellipse comprising two relatively movable parts each carrying a family of circular curves on it, the curves in each family having arithmetically increasing diameters with a common tangent point, the diameters of the curves corresponding to those of the other part, the parts being capable of being placed so that the families of curves are superposed with the centers of curvature of the circular curves on a cornmon axis and the families of curves offset by in relation to each other so that a series of points of intersection between curves corresponding to an ellipse can be registered.

2. An apparatus in accordance with claim 1 in which one of the parts is transparent.

3. An apparatus in accordance with claim 1 in which the parts are in the form of sheets.

5 6 4. An apparatus in accordance with claim 1 in which the stencil is in the form of transparent synthetic resin. both parts are in the form of transparent sheets. 9. An apparatus in accordance with claim 1 in which the 5. An apparatus in accordance with claim 1 in which tangent points are virtual. both the parts are in the form of plastic foil. n

6. A11 apparatus in accordance with claim 1 in which at r, References Clted least one of the parts is in the form of a stencil adapted UNITED STATES PATENTS for use with a writing instrument for transferring points of intersection onto an underlay.

7. An apparatus in accordance with claim 6 in which the stencil has slots corresponding to the circular curves, the 10 WILLIAM ,D M AR'FIN, JR Primary Examiner slots lying symmetrically to each side of an axis passing through al1 centers of curvature. US. C1. X.R.

v8. An apparatus in accordance with claim 6 in which 33-174; 35--34 2,703,454 3/1955 Haywood. 2,933,818 4/1960 Palmer.