Drawing mechanisms · Perpendicular slides

Trammel of Archimedes

Follow two shoes in perpendicular slots and move the pen along their rigid arm. Draw a slender ellipse, turn it into a circle, or place the pen on a shoe to collapse the path 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 link closure error $0$

Move & trace

Make the geometry draw.

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 is horizontal–vertical: right and up are positive, depth points toward you. Angles use degrees in the controls; internal calculations and CSV use radians, metres and seconds.

The blue shoe slides horizontally and the coral shoe vertically. The pen fraction starts at the horizontal shoe and reaches the vertical shoe at unity; larger values extend beyond it. The gold arm is rigid.

At the midpoint the pen draws a circle. At the vertical shoe the ellipse degenerates into a line and speed vanishes at its endpoints; curvature is then undefined. Constant orientation rate generally gives changing pen speed.

Playback traces one displayed phase interval and stops. The pale path is a complete geometric reference; coral ink shows phases up to the current cursor. Moving backward retraces this phase view rather than erasing a physical recording. CSV has 721 kinematic samples, not measured data.

Input motion is imposed. These models solve geometry and 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 outside scope.

Ideal perpendicular prismatic joints and a rigid arm. Arm orientation changes at an imposed constant rate, not a fixed motor shaft. No friction, mass, clearance, force or drive dynamics.

Start with the essentials

Focus question: Where must the pen sit to draw a circle rather than an ellipse?
One-sentence intuition: The two shoes supply perpendicular sinusoidal components; the pen location sets their amplitudes.

Core mathematical model

Two perpendicular shoes

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

Their pivot spacing stays fixed. The arm orientation is supplementary to the increasing parameter angle.

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.

Ellipse semiaxes

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

Equal nonzero semiaxes give a circle. A zero semiaxis gives 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

Angular rate is constant and positive. The pen is not generally moving uniformly.

Common difficulties

The reference is not a measurement

Typical misconceptionThe pale curve is an independent experimental check.

Better mental modelIt is the complete ideal model path. Link-closure residuals and independent geometric tests check consistency; real data are not acquired.

Motion is prescribed

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 require a separate analysis. The three-dimensional part thicknesses only explain assembly.

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 same path is traversed faster. Velocities scale with angular rate and accelerations with its square.

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