The Rytz’s axis construction is a basic method of
descriptive geometry to find the axes, the
semi-major axis and
semi-minor axis
In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter. The semi-major axis (major semiaxis) is the lo ...
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 (see
ellipse).
Rytz’s construction is a
classical construction of
Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...
, in which only
compass
A compass is a device that shows the cardinal directions used for navigation and geographic orientation. It commonly consists of a magnetized needle or other element, such as a compass card or compass rose, which can pivot to align itself wit ...
and
ruler
A ruler, sometimes called a rule, line gauge, or scale, is a device used in geometry and technical drawing, as well as the engineering and construction industries, to measure distances or draw straight lines.
Variants
Rulers have long ...
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
is: The tangents at the ellipse points of one diameter are parallel to the second diameter (see second diagram).
Problem statement and solution
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. These diameters of the ellipses are no more orthogonal but as images of orthogonal diameters of the circle they are
conjugate (the tangents at the end points of one diameter are parallel to the other diameter !). 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
(1) rotate point
around
by 90°.
(2) Determine the center
of the line segment
.
(3) Draw the line
and the circle with center
through
. Intersect the circle and the line. The intersection points are
.
(4) The lines
and
are the of the ellipse.
(5) The line segment
can be considered as a paperstrip of length
(see
ellipse) generating point
. Hence
and
are the . (If
then
is the semi- axis.)
(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
and
is still true.
Proof of the statement
The standard proof is performed geometrically.
[Ulrich Graf, Martin Barner: ''Darstellende Geometrie.'' Quelle & Meyer, Heidelberg 1961, , p.114] 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
(1): Any ellipse can be represented in a suitable coordinate system parametrically by
:
.
:Two points
lie on conjugate diameters if
(see
Ellipse: conjugate diameters.)
(2): Let be
and
:
:two points on conjugate diameters.
:Then
and the midpoint of line segment
is
.
(3): Line
has equation
:The intersection points of this line with the axes of the ellipse are
:
(4): Because of
the points
lie on the circle with center
and radius
:Hence
(5):
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 (4) and (5) are still valid and the configuration shows now the ''
2nd paper strip method'' (see diagram).
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
(the center) and
(two conjugate half-diameters) one is able to calculate points and to draw the ellipse.
If necessary: With
one gets the four vertices of the ellipse:
References
*
*
*
{{Reflist
External links
animation of Rytz's construction
Euclidean geometry
Descriptive geometry
Articles with example Python (programming language) code