# Trammel of Archimedes

A trammel of Archimedes is a mechanism that generates the shape of an ellipse. It consists of two shuttles which are confined ("trammelled") to perpendicular channels or rails and a rod which is attached to the shuttles by pivots at fixed positions along the rod. As the shuttles move back and forth, each along its channel, the end of the rod moves in an elliptical path. The semi-axes a and b of the ellipse have lengths equal to the distances from the end of the rod to each of the two pivots.

An ellipsograph is a trammel of Archimedes intended to draw, cut, or machine ellipses, e.g. in wood or other sheet materials. An ellipsograph has the appropriate instrument (pencil, knife, router, etc.) attached to the rod. Usually the distances a and b are adjustable, so that the size and shape of the ellipse can be varied.

The history of such ellipsographs is not certain, but they are believed to date back to Proclus and perhaps even to the time of Archimedes.

Wooden versions of the trammel of Archimedes have been produced also as toys or novelty items, and sold under the name of Kentucky do-nothings, nothing grinders, do nothing machines, or bullshit grinders. In these toys the drafting instrument is replaced by a crank handle, and the position of the sliding shuttles is usually fixed.

## Mathematics

Loci of some points along and beyond a trammel of Archimedes, the green circle being the loci of its midpoint – in the SVG file, move the pointer over the diagram to move the trammel

Let C be the outer end of the rod, and A, B be the pivots of the sliders. Let p and q be the distances from A to B and B to C, respectively. Let us assume that sliders A and B move along the y and x coordinate axes, respectively. When the rod makes an angle θ with the x-axis, the coordinates of point C are given by

$x=(p+q)\cos \theta \,$
$y=q\sin \theta \,$

These are in the form of the standard parametric equations for an ellipse in canonical position. The further equation

${\frac {x^{2}}{(p+q)^{2}}}+{\frac {y^{2}}{q^{2}}}=1$

is immediate as well.

The trammel of Archimedes is an example of a four-bar linkage with two sliders and two pivots, and is special case of the more general oblique trammel. The axes constraining the pivots do not have to be perpendicular and the points A, B and C can form a triangle. The resulting locus of C is still an ellipse.