3          Ellipse-Geometry

The ovalturner should know at least the following about ellipses from his school books.


                      Figure 2101 Ellipse axes and linear eccentricity
                                            a) Half-axes (semi-axes)
                                            b) Focuses (foci)

The ellipse is a conic section; one receives it, if one cuts through a circular cone or a circular cylinder diagonally.  The ellipse has two axes which are perpendicular to each other at the centre M (Figure 2101a). The half-axes have the length AM = a and BM = b. The half-axes are the two important parameters for ovalturning. They are given mostly.
To adjust the ovalturning lathe the

half-axes difference                   d = a – b

is the essential parameter, not to confound with the

axes ratio                                   β = b/a,

abbreviated by the letter 
β (beta). It plays a role with the adjustment of the Indexer.
Another measure is the

linear eccentricity                     e.

It is not needed for the adjustment of the ovalturning lathe, it is however necessary when drawing the ellipse using the thread construction (gardener construction). The linear eccentricity is the distance of the two foci F1 and F2 on the large axis from the centre M of the ellipse. One locates them by drawing a circular arc around B with the radius a (Figure 2101b) or with the formula:
                                                e = √(a²−b²).

The ellipse perimeter u is also important. It cannot be calculated accurately, so an approximation formula is usedas follows:

                                                u ≈π (1,5(a + b) - √ab )     or with     ß = b/a
                                                u ≈ a π(1,5(1 + ß) -√ß).

The area A of an ellipse is     A = abπ .

The simplest method of drawing an ellipse is the thread construction, e.g. for marking and cutting an elliptical blank from a board or a block. Draw the two axes AA and BB in the wanted lengths, with the intersection M, on the board, set a pencil  compass to the  half-axis length MA = a and draw an arc - as in Figure 2101b - around one of the end points B of the short axis. It cuts the large axis in the two foci F1 and F2. Fix nails into these points and also into one of the B points. Draw a thread loop around these 3 nails knotting at the extreme ends (Figure 2102a). Pull the nail from point B and draw the ellipse with a pencil. Keep the thread tight when drawing the ellipse (Figure 2102b).

 Thread construction of the Ellipse:

Figure 2102a  Thread ring around three nails

Figure 2102b  Drawing the ellipse

Ellipsographs are mechanisms, which guide a pencil or pen for drawing an ellipse onto a fixed drawing board. Instead of the pen a glass cutter or a knife can be used for cutting paper or cardboard as frame inlays. The ellipse axes can be adjusted on the ellipsograph. The kinematic principle of the ellipsograph is achieved by using double slider linkage, the trammel (Figure 2103). The sliders 2 and 4 run in stationary cross slot 3. They are connected in their joints A and B by a coupler 1, the point C of which writes the ellipse kC into the stationary plane. The adjustment of the required half axes is calculated by the distances BC = a and AC = b. The distance AB = a - b = d is the half-axes difference. The midpoint M of the distance AB runs along the circle kM around the ellipse centre M0. The radius of this circle is r = d/2.

     Figure 2103 Kinematic scheme of the Ellipsograph

Using the Ellipsographs shown in the following figures ellipses can be drawn and, if the pen/pencil is substituted by a knife, paper or card ellipses may be cut out. When cutting glass exchange the pen/pencil with a glass cutter.

Figure 2104 Ellipse- and Circle-Glas-Cutter (Firma  J. Bohle, D 42755 Hann)


     Figure 2105 Drawing-Ellipsograph
      (Gebrüder Haff GmbH, D 87459 Pfronten)




The ellipsograph in Figure 2106 draws and cuts ellipses to the side of the mechanism. This allows the smallest ellipses to be drawn. The mechanism corresponds with a patent by F. O. Kopp. Instructions for its use have been published [6.1].


Figure 2106 Kopp-Ellipsograph
(Workshop J. Volmer, Chemnitz 1999)