Rytz's construction

Using rytzschen axis construction, it is possible on the basis of two conjugated diameters of an ellipse to find its major and minor axis of the ellipse and its apex. When rytzschen axle construction is a classical construction of Euclidean geometry in which only ruler and compass are allowed as aids. The design is named after its creator David Rytz of Brugg, 1801-1868.

Problem

Figure 1 shows the given and required quantities. Given are the two conjugate diameters and (blue), sought the axes and the ellipse (red). For clarity, the corresponding ellipse is also drawn, she is however not given, nor is it a direct result of rytzschen axle construction. With ruler and compass can be constructed only single ellipse points, but not the entire ellipse. Method of drawing an ellipse usually put the axes of the ellipse as given ahead. Axes are not the only two conjugated diameter of an ellipse where the Rytzsche axle construction can be considered as pre-processing for displaying the ellipse.

Conjugate diameter

An ellipse can be considered as affine image of their main circuit under a vertical axis affinity. Figure 1 shows the ellipse next to their main circuit. The affine transformation, which is converted to indicated by dashed arrows. The archetype of an ellipse diameter below the figure is a circle of diameter. The defining characteristic of conjugate diameters of an ellipse and is that their archetypes and are perpendicular. In this sense, the axes of an ellipse specific conjugate diameters, in which not only their archetypes but the diameters are even perpendicular. The Rytzsche axle design applies to any two conjugate diameters conjugate diameter of those of the corresponding ellipse that are mutually perpendicular. These special conjugated diameter, the axes of the ellipse.

Construction

Figure 2 shows the steps of construction Rytzschen axis. Where are the bold blue marked conjugate diameter and which intersect at the center of the ellipse. From each conjugate diameter of an endpoint is selected on and on. The angle is either dull () as in the figure, or pointed (). Stood conjugated diameter perpendicular to each other (), the axes were already found, in this case, they would be identical to the given conjugate diameters.

In the first step is rotated about the center of in the direction. The result is the point. The points define the line. The midpoint of the segment is. The next step is to propose a circuit so that it passes through the center of the ellipse. The points of intersection of this circle with the line defining the points. and are chosen so that, seen from the point from the point on the same side as and is on the same side. You draw next from the point of two straight lines, one through and the other by. These lines intersect at a right angle ( as an associated Thales circle ).

The statement of Rytzschen axle construction is now that the ellipse axes on the straights by and respectively, and lie, and that the length of the track of the length of the ellipse major vertex and the length of the ellipse corresponds to the vertex addition. In the last step we therefore proposes two circles around with the radii and. To find the main peaks and the distance on the straight line passing through the side and corresponding to the vertices and at a distance of on the straight through.

Justification of the construction

The Rytzsche axis design is based on the properties of conjugated ellipse diameter and is not understood without consideration of the ellipse as affine loop display. This section gives an insight into the background of the design and makes it possible to understand their correctness. Reading this portion is not necessary to carry out the construction. This section assumes that in addition to the given conjugate diameters of the result of the construction ( the ellipse axes and vertices ) is already known to the basis of which to make the design steps insightful.

The archetypes of the conjugate diameter

The ellipse, the conjugate diameter, and are given, can be considered as affine image of their main circle with respect to an affine transformation. Figure 3 shows the ellipse with its main circuit and its secondary circuit. The points are endpoints of or which intersect at the center of the main circle. The Archetypes and (green) and of respect are thus circle diameter of the ellipse major circle. Because of the property that and conjugate diameters, their archetypes and are perpendicular. The archetype of or with respect to the corresponding endpoints or the circle diameter or. The intersection points of the circle diameter and the secondary circuit of the ellipse are the points or.

At the beginning of the construction are only the points, and, where. Neither the archetypes and the conjugate diameter, yet the points, and are well known, yet they are determined in the course of construction. It is important only for the understanding of the construction. If these points are referred to in the further course of the description is to be understood as " If these points were known, then one would find that ... ".

Parallels to the ellipse axes

Interestingly, the lines and parallel to the axes of the ellipse, and therefore form a right angle. The same applies to the routes and on point. This can be explained as follows: The affine transformation that maps the ellipse major circle on the ellipse, the minor axis of the ellipse as Fixgerade. Since a straight line through a point (for example) and its image point (for example ) is also a Fixgerade must due to the parallel loyalty affine images, the line through and be a parallel to the minor axis. The same argument applies for the line through and. To indicate that the lines are parallel to and through and or ellipse major axis to the ellipse considered as their affinity image side circuit and the argument used accordingly.

The recognition that lines and are parallel to the axes of this help is not on, because the points are not known. The next step uses these parallels, however, sent out to still find the axes.

Find the ellipse axes

If one turns, as shown in Figure 3, the ellipse diameter together with its archetype to about the center in the direction, so comes with the cover and the rotated point coincides with and related to. The point goes over. Due to the parallelism of the ellipse and with an axis, and the parallelism with the other axis of the ellipse forming the dots, and a rectangle. From this square, however, are only the points and known. This is sufficient, however, to find its diagonal intersection.

The diagonal intersection is obtained by halving the diagonal. The other is located on the diagonal line passing through, and ( because of the diagonal intersection point and must lie on the diagonal diameter of the main circuit ), however, have their end points, and yet to be identified through the structure. Important to finding the ellipse axes but is merely that the ellipse major axis is parallel to and through the ellipsis minor axis is parallel to a corresponding through.

If one extends the already known diagonal as shown in Figure 4, it intersects the ellipse major axis at a point and the ellipse minor axis in, and there are isosceles triangles and ( share the diagonals a square into four isosceles triangles, plus set of beams ) in. The same is true for the triangles and. This property is utilized for the construction of the points: since the length of the path equal to the length of the lines and must be to find and the intersection points with a circle of radius. With the points and is now the location of the ellipse axes known ( on the straights by and respectively ). There is simply no other vertices.

Identification of the ellipse vertex

The length of the major axis corresponding to the length of the radius is the main circuit. The length of the minor axis is equal to the radius of the secondary circuit. The radius of the main circuit, however, is equal to the length of the distance and the radius of the auxiliary circle is gleichder length of the route. For the determination of and the location of points and are not designed must because the following identities hold:

In the construction, ie, the length of the ellipse axes can already read: and. With this information, can be the main and secondary circuit of the ellipse draw. The principal vertex and can be found as the intersection points of the main circle with the ellipse major axis. The decision on which of the two axes found it is the main or the secondary axis, due to the following: is the image of with respect to the affine transformation that maps the ellipse major circle on the ellipse. As it is, and is a contraction in the direction of the major axis, the major axis on the opposite side of, and thus must be located running through the point situated on the side of the non-rotated elliptical diameter. This is independent of the initial choice of the points. All that matters is that is turned off when twisted around, since only then the point is on the archetype of the conjugate diameter. The ellipse principal axis is then made ​​by always viewed on the opposite side of.