Archimedes' Trammel
The ellipsograph: a mechanical linkage where two sliders constrained to perpendicular tracks drive a pencil point that traces a perfect ellipse. Drag the pencil to steer, or let it animate. Adjust the rod lengths to reshape the curve from a circle to a razor-thin line.
About this lab
The trammel of Archimedes is a mechanism that constrains a point to trace an ellipse. It consists of a rigid rod with two sliders (shuttle blocks) that are confined to perpendicular grooves. As the rod rotates, the sliders move back and forth in their respective channels, and any fixed point on the rod — not between the sliders — traces an elliptical path. If the point lies at one of the sliders, it traces a straight line; if the two slider-to-point distances are equal, it traces a circle. All other positions yield ellipses with varying eccentricity.
Mathematically, if slider A moves along the x-axis and slider B along the y-axis, and the distances from the pencil point P to sliders A and B are a and b respectively, then as the rod sweeps through angle theta, the coordinates of P are x = a cos(theta) and y = b sin(theta) — the standard parametric equations of an ellipse with semi-axes a and b. The eccentricity e = sqrt(1 - (b/a)^2) when a > b. This elegant relationship between constrained linear motion and elliptical curves was known in antiquity and remains a staple in mechanical engineering courses.
Trammel mechanisms have been used historically to cut elliptical shapes in woodworking (the "elliptic trammel" or "ellipsograph" jig), to machine elliptical ports in engine cylinders, and in various decorative arts. The device is sometimes sold as a mathematical toy under the name "do-nothing machine" or "bullshit grinder" — a playful name for what is in fact a beautiful demonstration of how simple constraints produce complex curves. The same kinematics appear in Cardan gear mechanisms and in the coupling rods of some steam engines.