Rytz's construction

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Using the Rytz’s axis construction, it is possible to find the major and minor axis and the vertices of an ellipse, starting from two conjugated diameters. Rytz’s construction is a classical construction of Euclidean geometry, in which only compass and ruler are allowed as aids. The design is named after its inventor David Rytz of Brugg, 1801–1868.

Problem statement[edit]

Figure 1: Given sizes and results

Figure 1 shows the given and required quantities. The two conjugate diameters d_1', and d_2' (blue) are given, and the axes a and b of the ellipse (red) are required. For clarity, the corresponding ellipse e is also shown, however, it is neither given, nor is it a direct result of Rytz's construction. With ruler and compass only a few points of the ellipse can constructed, but not the entire ellipse. Methods of drawing an ellipse usually require the axes of the ellipse to be known.

Conjugate diameters[edit]

An ellipse can be seen as an image of the unit circle under an affine transformation.

Figure 1 shows the ellipse e next to the unit circle k_h. The affine image \alpha, which transforms the unit circle k_h into the ellipse e is indicated by the dashed arrows. The preimage of an ellipse diameter under the image \alpha is a circle of diameter k_h.

Construction[edit]

Figure 2: Construction

Figure 2 shows the steps of the Rytz’s construction. The conjugated diameters d_1' and d_2' (thick blue lines) are given, which meet at the center M of the ellipse. A point on each conjugate diameter is selected: U' on d_1' and V' on d_2'. The angle \angle(U' M V') is either obtuse (> 90^\circ) as shown in the figure, or acute (< 90^\circ). If the conjugated diameters are standing perpendicular to each other (= 90^\circ), the axes of the ellipse are already found: In this case, they are identical to the given conjugated diameters.

In the first step, the point U' is rotated 90^\circ around the center M toward point V'. The result is the point U'_r. The points U'_r and V' define the line g. The midpoint of the line \overline{U'_r V'} is S. The next step is drawing a circle t around S so that it passes through the center M of the ellipse. The intersections of the circle with the line g define the points R and L. R and L are selected such that R is located on the same side as U'_r and L is located on the same side as V', as viewed from the point S. Next, you draw from the point M two straight lines, one through R and the other through L. These lines intersect M at a right angle (as Thales' theorem states).

The proposition of the Rytz’s construction is that the directions of the ellipse axes are indicated by the vectors \overline{M L} and \overline{M R}, and the length of the line \overline{V 'R} is the length of the ellipse’s major axis and the length of the \overline{V 'L} corresponds to the length of the ellipse’s minor axis. In the last step we therefore propose two circles around M with the radii a and b. The major vertices S_1 and S_2 are at a distance a of M on the line through L and the minor vertices S_3 and S_4 are at a distance b of M on the line through R.

Algorithm[edit]

The following Python code implements the algorithm described by the construction building steps.

References[edit]

  • Rudolf Fucke, Konrad Kirch, Heinz Nickel (2007). Darstellende Geometrie für Ingenieure [Descriptive geometry for engineers] (in German) (17th ed.). München: Carl Hanser. p. 183. ISBN 3446411437. Retrieved 2013-05-31. 
  • Klaus Ulshöfer, Dietrich Tilp (2010). "5: Ellipse als orthogonal-affines Bild des Hauptkreises" [5: "Ellipse as the orthogonal affine image of the unit circle"]. Darstellende Geometrie in systematischen Beispielen [Descriptive geometry in systematic collection of examples]. Übungen für die gymnasiale Oberstufe (in German) (1st ed.). Bamberg: C. C. Buchner. ISBN 978 3 7661 6092 8.