# Rytz's construction

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

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

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

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

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

## References

• 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.