Drawing mechanisms · Perpendicular slides

Trammel of Archimedes

On a trammel of Archimedes, where should the pen sit to draw a circle? At the midpoint of the arm, halfway between the two shoes. Each shoe slides back and forth in its own slot, and a pen on the arm combines the two motions: its sideways swing equals its distance from the shoe in the vertical slot, and its up-down swing equals its distance from the other shoe. Halfway, the swings are equal and the ellipse becomes a circle. A pen on a shoe draws a straight line.

Two shoes slide in perpendicular slots, joined by a rigid arm. Move the pen along the arm to draw a slender ellipse or turn it into a circle, or put the pen on a shoe and watch the path collapse into a line.

Pen horizontal position $0$

Pen vertical position $0$

Pen speed $0$

Pen acceleration magnitude $0$

Shoe pivot distance $0$

Path eccentricity $0$

Local path curvature $0$

Largest constraint residual $0$

Move & trace

Let the linkage draw the curve

Orientation parameter

\theta

250.00\,{}^\circ

Prescribed angular rate

\omega

60.00\,{}^\circ/\mathrm s

Distance between shoe pivots

L

1.60\,\mathrm m

Pen position along the arm

t

0.35\,

The drawing plane holds the horizontal and vertical directions, with right and up positive and depth pointing toward you. The controls give angles in degrees; the calculations and the CSV use radians, metres and seconds.

The blue shoe slides horizontally and the coral shoe vertically. The pen fraction is zero at the horizontal shoe and one at the vertical shoe; larger values put the pen beyond it. The gold arm is rigid.

With the pen at the midpoint, it draws a circle. With the pen at the vertical shoe, the ellipse collapses into a straight line, the speed drops to zero at its ends and the curvature there is undefined. Turning the arm at a constant rate generally does not move the pen at constant speed.

Playback traces the displayed phase range once and stops. The pale path is the full geometric reference, and the coral ink shows the phases up to the current cursor. Dragging backward simply redraws this view; it does not erase a physical recording. The CSV holds 721 kinematic samples, not measured data.

The input motion is prescribed. These models solve the geometry and its derivatives, not contact forces or a motor drive. Parts, pen supports and clearances are schematic; material penetration, inertia, friction, backlash, tolerances and structural safety are not modeled.

The two sliding joints are ideal and perpendicular, and the arm is rigid. The arm turns at a constant rate that is imposed directly, not by a motor on a fixed shaft. Friction, mass, clearance, forces and drive dynamics are not modeled.

Start with the essentials

Focus question: Where must the pen sit to draw a circle instead of an ellipse?
One-sentence intuition: The two shoes give the pen two perpendicular sinusoidal components, and where the pen sits on the arm sets their amplitudes.

Core mathematical model

Positions of the two shoes

\mathbf A=(L\cos\theta,0),\quad\mathbf B=(0,L\sin\theta)

The distance between their pivots stays fixed. The arm's angle is the supplement of the parameter angle, which keeps increasing.

A pen fixed along the arm

\mathbf Q=(1-t)\mathbf A+t\mathbf B

The fraction is measured from the horizontal shoe toward the vertical shoe.

Semiaxes of the ellipse

a=L|1-t|,\quad b=Lt

When the two semiaxes are equal and nonzero, the curve is a circle. When one semiaxis is zero, the curve is a line, not a regular ellipse.

Velocity and acceleration

\dot{\mathbf Q}=L\omega(-(1-t)\sin\theta,t\cos\theta),\quad\ddot{\mathbf Q}=-\omega^2\mathbf Q

The angular rate is constant and positive, but the pen's speed is generally not constant.

Common difficulties

The pale curve is not a measurement

Typical misconceptionThe pale curve is an independent experimental check.

Better mental modelIt is the full path of the ideal model. The constraint residuals and independent geometric tests check that the model is consistent; no real data are acquired.

The motion is set directly

Typical misconceptionA smooth trace proves that a real mechanism can be driven safely at this rate.

Better mental modelMass, friction, contact forces, clearances and drive dynamics need a separate analysis. The thicknesses of the three-dimensional parts only show how the pieces are assembled.

Run the experiment

01
Trace the default path
Use the drawing-plane view, then play or drag the coral handle.
What to observe: Each shoe stays in its slot while the pen follows an ellipse.
02
Move the pen
Compare the midpoint, slender and shoe presets.
What to observe: The midpoint draws a circle; the shoe draws a line.
03
Separate geometry from speed
Increase the angular rate and export the phase sweep.
What to observe: The pen covers the same path faster. Velocities grow in proportion to the angular rate, and accelerations grow with its square.

Geometry & apparatus: Il Giardino di Archimede · Peaucellier · Caltech · Elliptical Trammel