Rytz's construction


The Rytz’s axis construction is a basic method of descriptive geometry to find the axes, the semi-major axis and semi-minor axis and the vertices of an ellipse, starting from two conjugated half-diameters. If the center and the semi axis of an ellipse are determined the ellipse can be drawn using an ellipsograph or by hand.
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.
Conjugate diameters appear always if a circle or an ellipse is projected parallelly as images of orthogonal diameters of a circle or as images of the axes of an ellipse. An essential property of two conjugate diameters is: The tangents at the ellipse points of one diameter are parallel to the second diameter.

Problem statement and solution

The parallel projection of a circle that is in general an ellipse. A fundamental task in descriptive geometry is to draw such an image of a circle. The diagram shows a military projection of a cube with 3 circles on 3 faces of the cube. The image plane for a military projection is horizontal. That means the circle on the top appears in its true shape. The images of the circles at the other two faces are obviously ellipses with unknown axes. But one recognizes in any case the images of two orthogonal diameters of the circles. These diameters of the ellipses are no more orthogonal but as images of orthogonal diameters of the circle they are conjugate. This is a standard situation in descriptive geometry:
  • From an ellipse the center and two points on two conjugate diameters are known.
  • Task: find the axes and semi-axes of the ellipse.

Steps of the construction

rotate point around by 90°.

Determine the center of the line segment.

Draw the line and the circle with center through. Intersect the circle and the line. The intersection points are.

The lines and are the of the ellipse.

The line segment can be considered as a paperstrip of length generating point. Hence and are the.

The vertices and co-vertices are known and the ellipse can be drawn by one of the drawing methods.
If one performs a left turn of point, then the configuration shows the 2nd paper strip method and and is still true.

Proof of the statement

The standard proof is performed geometrically. An alternative proof uses analytic geometry:
The proof is done, if one is able to show that
  • the intersection points of the line with the axes of the ellipse lie on the circle through with center, hence and, and

Proof

: Any ellipse can be represented in a suitable coordinate system parametrically by
: Let be and

: Line has equation

: Because of the points lie on the circle with center and radius

:
The proof uses a right turn of point, which leads to a diagram showing the 1st paper strip method.

Variations

If one performs a left turn of point, then results and are still valid and the configuration shows now the 2nd paper strip method.

If one uses, then the construction and proof work either.

Computer aided solution

To find the vertices of the ellipse with help of a computer,
  • the coordinates of the three points have to be known.
A straight forward idea is: One can write a program that performs the steps described above. A better idea is to use the representation of an arbitrary ellipse parametrically:
With and one is able to calculate points and to draw the ellipse.
If necessary: With one gets the four vertices of the ellipse: