matematicas visuales visual math

In this page we study an instrument for drawing ellipses. It is called "Ellipsograph" or "Trammel of Archimedes".

The word 'trammel' in this context has a meaning of restrain, something impeding activity, a constraint that restrics freedom. The origin of this word is related with a net to catch fish.

We can see a classical representation of this tool in Cundy and Rollet's book 'Mathematical Models'.

Trammel of Archimedes, Ellipsograph: Cundy and Rollet image | matematicasVisuales
Classical representation of this tool in Cundy and Rollet's book 'Mathematical Models'.

"Four triangular cheek-pieces are bolted firmly to a base-plate which rests on the paper. They form the walls of two slots at right angles in which the sliders A and B can run. PAB is a slotted arm which can be screwed to the sliders at A and B in such a way that it is free to rotate on the slider, but not to slip along the slot AB. The sliders must be longer than the width of the slots at O to ensure free travel across the opening.(...) By adjusting the screws, ellipses of different sizes can be drawn by a tracing point at P." (Cundy and Rollet, p. 240)

We can say that the two sliders are 'trammelled' or confined by the two slots and the rod.

We want to prove that point P draws an ellipse.

Trammel of Archimedes, Ellipsograph: drawing ellipses | matematicasVisuales

Some properties of ellipses have been studied in other pages of matemáticasVisuales.

Ellipse and its foci
Every ellipse has two foci and if we add the distance between a point on the ellipse and these two foci we get a constant.
Equation of an ellipse
Transforming a circle we can get an ellipse (as Archimedes did to calculate its area). From the equation of a circle we can deduce the equation of an ellipse.
Archimedes and the area of an ellipse: an intuitive approach
In his book 'On Conoids and Spheroids', Archimedes calculated the area of an ellipse. We can see an intuitive approach to Archimedes' ideas.

To calculate coordinates of point P we are going to use this notation and trigonometry:

Trammel of Archimedes, Ellipsograph: notation | matematicasVisuales
Trammel of Archimedes, Ellipsograph: notation | matematicasVisuales

Then the X coordinate of point P is:

Trammel of Archimedes, Ellipsograph: X coordinate point P | matematicasVisuales

And the Y coordinate is:

Trammel of Archimedes, Ellipsograph: Y coordinate point P | matematicasVisuales

Then, the coordinates of point P are:

Trammel of Archimedes, Ellipsograph: coordinates point P drawing ellipse | matematicasVisuales

These coordinates verify the implicit formula of the ellipse:

Trammel of Archimedes, Ellipsograph: implicit equation ellipse | matematicasVisuales

The two semi-axis of the ellipse are a and b, it is to say, the distances between the end of the rod and the two pivoted points:

Trammel of Archimedes, Ellipsograph: semi-axis ellipse | matematicasVisuales
Trammel of Archimedes, Ellipsograph: semi-axis ellipse | matematicasVisuales

Changing the distances between points we get different ellipses:

Trammel of Archimedes, Ellipsograph: Changing the distances between points we get different ellipses | matematicasVisuales

This idea was used to build devices for drawing ellipses (called ellipsographs).

This instrument is in the Museo Nacional de Ciencia y Tecnología in Madrid:

Trammel of Archimedes, Ellipsograph: Ellipsograph in the Museo Nacional de Ciencia y Tecnología Madrid | matematicasVisuales
Ellipsograph in Museo Nacional de Ciencia y Tecnología in Madrid.

Each point in the rod draws an ellipse, as you can see in the next interactive application:

Trammel of Archimedes, Ellipsograph: Each point in the rod draws an ellipse | matematicasVisuales

You can get ellipses even if the point P is between the two sliding points:

Trammel of Archimedes, Ellipsograph: You can get ellipses even if the point P is between the two sliding points | matematicasVisuales
Trammel of Archimedes, Ellipsograph:  | matematicasVisuales

A circle is a particular case of ellipse:

Trammel of Archimedes, Ellipsograph: circles are particular cases | matematicasVisuales

REFERENCES

Tom Apostol and Mamikon Mnatsakanian, 'New Horizons in Geometry' (Chapter 9. Trammels, ), Mathematical Association of America, 2012.
H.Martin Cundy and A.P. Rollet, 'Mathematical Models', Oxford University Press, Second Edition, 1961.
Hilbert and Cohn-Vossen, Geometry and the Imagination. Chelsea Publishing Company. pag.278.
Robert C. Yates, 'A Handbook on curves and their properties', J.W.Edwards-Ann Arbor, 1947.
J.L. Coolidge, The Mathematics of great Amateurs. Second Edition. Claredon Press. Oxford. Jan de Witt's proof in page 124.
Historical Mechanisms for Drawing Curves by Daina Taimina. With De Witt's proof why the trammel describes an ellipse.

MORE LINKS

The Astroid as envelope of segments and ellipses
The Astroid is the envelope of a segment of constant length moving with its ends upon two perpendicular lines. It is also the envelope of a family of ellipses, the sum of whose axes is constant.
The Astroid is a hypocyclioid
The Astroid is a particular case of a family of curves called hypocycloids.
Ellipse and its foci
Every ellipse has two foci and if we add the distance between a point on the ellipse and these two foci we get a constant.
Archimedes and the area of an ellipse: an intuitive approach
In his book 'On Conoids and Spheroids', Archimedes calculated the area of an ellipse. We can see an intuitive approach to Archimedes' ideas.
Archimedes and the area of an ellipse: Demonstration
In his book 'On Conoids and Spheroids', Archimedes calculated the area of an ellipse. It si a good example of a rigorous proof using a double reductio ad absurdum.
Plane developments of geometric bodies (4): Cylinders cut by an oblique plane
We study different cylinders cut by an oblique plane. The section that we get is an ellipse.
Plane developments of geometric bodies (8): Cones cut by an oblique plane
Plane developments of cones cut by an oblique plane. The section is an ellipse.
Plane developments of geometric bodies (2): Prisms cut by an oblique plane
Plane nets of prisms with a regular base with different side number cut by an oblique plane.