---
title: "Peaucellier Inversor Interactive Lab | Sandbox Physics"
description: "A straight line. Seven moving links. No straight guide. How can hinged bars draw a straight line without a guide?"
canonical_url: "https://sandboxphysics.com/labs/peaucellier-inversor/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/peaucellier-inversor/"
last_modified: "2026-10-21"
---

Drawing mechanisms · Geometric inversion

# Peaucellier Inversor

Swing a circular crank and watch a hinged rhombus guide the pen along a straight line. Change the circle to bend the path, inspect the stacked pivots, and test the geometric invariant behind the drawing.

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

Ideal planar rigid links with prescribed crank angle. Only a nonsingular working arc is shown. Thicknesses and pin stacking are schematic; no friction, inertia, collision, contact load or tolerance model.

Physics tutorial

## A straight path made entirely of hinges

**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**: What must be true of the input circle for the output to be straight?
- **One-sentence intuition**: The rhombus keeps the input and pen on the same radial line, with a constant distance product.

### Core mathematical model

#### Geometric inversion

$$
K=\ell^2-r^2,\quad\mathbf Q=\frac{K\mathbf P}{|\mathbf P|^2}
$$

The fixed pivot is the origin; two long links and four equal rhombus sides impose this relation.

#### The input circle

$$
\mathbf P=(D+R\cos\theta,R\sin\theta)
$$

Its center is the other fixed pivot. Equal radius and pivot distance put the origin on the circle.

#### The exact straight path

$$
R=D\quad\Longrightarrow\quad Q_x=\frac K{2D},\quad Q_y=\frac K{2D}\tan\frac\theta2
$$

Only the safe crank arc is shown. Extending the crank through the inversion center is singular.

#### The more general output circle

$$
(D^2-R^2)|\mathbf Q|^2-2KDQ_x+K^2=0
$$

The same constraint becomes a line when the quadratic coefficient vanishes.

### 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:** Every bar keeps its length while the unguided pen follows a vertical line.

2. 02

   #### Break the straight path

   Change the crank radius ratio, keeping the two fixed pivots in place.

   **What to observe:** The pen follows a circular arc even though the inversion product remains constant.

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)