---
title: "Trammel of Archimedes Interactive Lab | Sandbox Physics"
description: "Two straight slides draw an ellipse. Where should the pen sit to draw a circle?"
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language: "en"
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last_modified: "2026-10-21"
---

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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

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.

Physics tutorial

## A rigid arm makes a family of curves

**Background** A mechanism can encode a geometric construction in its constraints.

**Why it matters** Follow the pen and then test the rods that constrain it.

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 misconception** The pale curve is an independent experimental check.

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

#### Motion is prescribed

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

**Better mental model** Mass, friction, contact forces, clearances and drive dynamics require a separate analysis. The three-dimensional part thicknesses only explain assembly.

### Run the experiment

1. 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.

2. 02

   #### Move the pen

   Compare the midpoint, slender and shoe presets.

   **What to observe:** The midpoint draws a circle; the shoe draws a line.

3. 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.

Physical conclusion

The visible trace and velocity plots come from the same linkage coordinates.

Geometry & apparatus: [Il Giardino di Archimede · Peaucellier](https://archimede.dimai.unifi.it/archimede/archimede_NEW_inglese/curve/geomeccan0.php?id=5) · [Caltech · Elliptical Trammel](https://robotics.caltech.edu/wiki/images/4/47/EllipticalTrammel.pdf)

Motion & dynamics · Starter path

## From a falling ball to an orbit

Measure gravity, split motion into components, add contact forces and collisions, then carry the same laws into orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)