Rytzsche axis construction

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Rytz construction: beginning - end

The Rytz axis construction is a method of descriptive geometry named after its Swiss inventor David Rytz , to construct the vertices and semiaxes of an ellipse with the help of compasses and ruler, if the center and two points on two conjugate diameters are known. After the vertices have been constructed, the ellipse can then be drawn by hand or with an elliptical compass using one of the numerous methods .

Cube with circles in bird's eye view
Rytz construction in 6 steps.
The center M and two conjugate radius MP, MQ of an ellipse are given.
Wanted: the vertex of the ellipse.

Description of the problem and its solution

In the case of a parallel projection of an ellipse or a circle, which is customary in descriptive geometry , the main axes or two orthogonal diameters of the circle are mapped onto conjugate diameters of the image ellipse.

  • Two diameters of an ellipse are called conjugate if the tangents at the points of one diameter are parallel to the other diameter. For a circle , two diameters are conjugate if they are orthogonal .

The illustration opposite shows a bird's eye view (oblique parallel projection) of a cube with circles. The upper (horizontal) side of the cube is shown undistorted from a bird's eye view (the image panel is horizontal). So the image of the top circle is again a circle. The other two circles are mapped onto ellipses, of which two conjugate diameters (images of perpendicular diameters of the circles) can easily be constructed. But they are not the main axes of the image ellipses. This is a standard situation in descriptive geometry:

  • The center point and two points on two conjugate diameters of an ellipse are known.
  • Task : Construct the main axes and vertices of the picture ellipse.
Construction steps

(1) Turn point to 90 °. (2) Find the midpoint of the line . (3) Draw the straight line and the circle with the center point . Cut the circle with the straight line. The intersections are . (4) The straight lines and are the axes of the ellipse. (5) The segment can be understood as a paper strip of the length with which the ellipse point is generated. So and are the semi-axes of the ellipse. (If is, is the semi-major axis .) (6) With this, the vertices of the ellipse are also known and the ellipse can be drawn with one of the methods .




If the first step is to turn to the left with the same specification , the configuration of the second paper strip method results and is also valid here.

Proof of the method

To prove the method

The standard proof is given geometrically (see below). The analytical proof is easier to understand:

The proof is given when it can be shown that:

  • The points of intersection of the straight line with the axes of the ellipse lie on the circle through with the center point . So is and and
proof

(1): Every ellipse can be represented in a suitable coordinate system by a

Parameter representation are described.
Two points lie on conjugate diameters if is. (see conjugate diameter )

(2): Let it be and

two points on conjugate diameters.
Then , and the center of the track is .

(3): The straight line has the equation

The intersections of this straight line with the axes of the ellipse are
Rytz: Left turn of the point

(4): Because of the points are on the

Circle with center and radius
So is

(5):

The proof uses a clockwise rotation of the point , resulting in a configuration like the 1st paper strip method .

Variations

Leads to a left turn of the point by, the results (4) and (5) are still valid, and the configuration shows the 2nd paper strip method (see Fig.). If one uses , the construction and the proof are still valid.

Solution with the help of a computer

To find the vertex of an ellipse with the help of a computer you have to

  • the coordinates of the points must be known.

One can try to write a program that mathematically reproduces the above construction steps. A more effective method uses the parametric representation of any ellipse

Where the midpoint are and (two conjugate radiuses). This enables you to calculate any number of points and draw the ellipse as a polygon .

If necessary: ​​With you get the 4 vertices of the ellipse: (see ellipse )

Geometric proof of the method

Figure 1: Given sizes and results

An ellipse can be viewed as an affine image of its main circle under a perpendicular axis affinity . In addition to the ellipse, Figure 1 shows its main circle . The affine mapping which is converted into is indicated by dashed arrows. The archetype of an ellipse diameter below the figure is a circle diameter of . The defining property of conjugate diameters and an ellipse is that their archetypes and are perpendicular to each other.

The archetypes of the conjugate diameters

Figure 3: Archetypes of the conjugate diameters

The ellipse, whose conjugate diameter and are given, can be viewed as an affine image of its main circle with respect to an affine map . Figure 3 shows the ellipse with its main circle and its secondary circle . Let the points and be end points of or that intersect in the center of the main circle. The archetypes and (green) of and regarding are thus the diameter of the main ellipse circle . Due to the property that and are conjugate diameters, their archetypes and are perpendicular to each other. The archetype of or in relation to are the corresponding end points or the circle diameter or . The points of intersection of the circle diameter or with the secondary circle of the ellipse are the points or .

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

Parallels to the axes of the ellipse

Figure 4: Parallels to the ellipse axes

Interestingly, the lines and are parallel to the axes of the ellipse and therefore form a right angle in . The same applies to the routes and in the point . This can be explained as follows: The affine mapping , which maps the main circle of the ellipse onto the ellipse, has the minor axis of the ellipse as a fixed line . Since a straight line through a point (for example ) and its image point (for example ) is also a fixed line, the straight line through and must be parallel to the minor axis due to the parallels affine mapping . The same argument applies to the straight line through and . In order to show that the straight lines through and or and are parallel to the main axis of the ellipse, one considers the ellipse as an affine image of its minor circle and applies the argument accordingly.

The knowledge that the lines and lie parallel to the searched axes does not help because the points and are not known. The following step uses these parallelism cleverly to find the axes anyway.

Finding the axes of the ellipse

Figure 5: Finding the ellipse axes

By turning, as shown in Figure 3, the elliptical diameter , together with its prototype to about the center in the direction , so the pre-images join , and to cover, and the rotated point coincides with and with together. The point passes into . Because of the parallelism of and with an axis of the ellipse and the parallelism of and with the other axis of the ellipse, the points are , , and a rectangle as seen in Figure 4. Of this rectangle, however, only the points and are known. But this is sufficient to find its diagonal intersection.

The diagonal intersection is obtained by halving the diagonal . The other diagonal lies on the straight line through and (because the diagonal intersection is and the diagonal must lie on a diameter of the main circle), but its end points and the construction have not yet been identified. In order to find the ellipse axes, it is only important that the main axis of the ellipse is parallel to through and correspondingly the minor axis of the ellipse is parallel to through .

If the already known diagonal is lengthened as shown in Figure 5, it intersects the main axis of the ellipse in one point and the minor axis of the ellipse in , resulting in isosceles triangles and in (the diagonals divide a rectangle into four isosceles triangles, plus the principle of rays). The same applies to the triangles and . This property is used for the construction of the points and : Since the length of the line must be equal to the length of the line or , one finds or as intersection points of a circle around with radius . With the points and the position of the elliptical axes is now known (on the straight lines through and or ). Only the vertices are missing.

Identification of the ellipse vertices

Figure 6: Identification of the vertices

The length of the main axis corresponds to the length of the radius of the main circle. The length of the minor axis is equal to the radius of the minor circle. But the radius of the main circle is equal to the length of the line and the radius of the secondary circle is equal to the length of the line . To determine and , the position of the points and must not be constructed, since the following identities apply:

The length of the ellipse axes can already be read off in the construction: and . With this information, the main and secondary circles of the ellipse can be drawn. The main vertices and are found as the intersection points of the main circle with the main axis of the ellipse. The decision as to which of the two axes found is the main or the minor axis is justified as follows: is the image of with respect to the affine mapping that maps the main ellipse circle onto the ellipse. Since it is a contraction in the direction of the major axis, the major axis must be on the opposite side of and therefore pass through the point that is on the side of the unrotated ellipse diameter . This is independent of the initial choice of points and . The only decisive factor is that in the rotation on is turned off because the only point on the prototype of the conjugated diameter is. The main axis of the ellipse is then always on the opposite side of .

literature

  • Rudolf Fucke, Konrad Kirch, Heinz Nickel: Descriptive geometry for engineers . 17th edition. Carl Hanser, Munich 2007, ISBN 3-446-41143-7 , p. 183 ( online at google-books [accessed on May 31, 2013]).
  • Klaus Ulshöfer, Dietrich Tilp: Descriptive geometry in systematic examples . 1st edition. CC Buchner, Bamberg 2010, ISBN 978-3-7661-6092-8 , 5: Ellipse as an orthogonal-affine image of the main circle (exercises for the upper school level).

Web links

Individual evidence

  1. ^ Ulrich Graf, Martin Barner: Descriptive Geometry. Quelle & Meyer, Heidelberg 1961, ISBN 3-494-00488-9 , p.114