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.

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).

Conjugate diameters appear always if a circle or an ellipse is projected parallelly (the rays are parallel) as images of orthogonal diameters of a circle (see second diagram) or as images of the axes of an ellipse.

An essential property of two conjugate diameters

The parallel projection (skew or orthographic) of a circle that is in general an ellipse (the special case of a line segment as image is omitted).

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 (as circle).

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.

This is a standard situation in descriptive geometry: (1) rotate point

Intersect the circle and the line.

can be considered as a paperstrip of length

(see ellipse) generating point

(6) 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 (see second diagram in next section) and

The standard proof is performed geometrically.

[1] An alternative proof uses analytic geometry: The proof is done, if one is able to show that (1): Any ellipse can be represented in a suitable coordinate system parametrically by (2): Let be

lie on the circle with center

The proof uses a right turn of point

, which leads to a diagram showing the 1st paper strip method.

If one performs a left turn of point

, then results (4) and (5) are still valid and the configuration shows now the 2nd paper strip method (see diagram).

, then the construction and proof work either.

To find the vertices of the ellipse with help of a computer, 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

(two conjugate half-diameters) one is able to calculate points and to draw the ellipse.