---
title: "Pantograph Drawing Copier Interactive Lab | Sandbox Physics"
description: "How can four bars copy a drawing at a different size? The hinged parallelogram keeps the fixed pivot, tracer and pen on one line, with a constant ratio of their distances from the pivot. Every point of the drawing is copied at that ratio, so the shape stays the same. Doubling the scale doubles lengths and pen speed, and makes each enclosed area four times as large."
canonical_url: "https://sandboxphysics.com/labs/pantograph-copier/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/pantograph-copier/"
last_modified: "2026-10-23"
---

Linkage geometry · Similarity

# Pantograph Drawing Copier

**How can four bars copy a drawing at a different size?** The hinged parallelogram keeps the fixed pivot, tracer and pen on one line, with a constant ratio of their distances from the pivot. Every point of the drawing is copied at that ratio, so the shape stays the same. Doubling the scale doubles lengths and pen speed, and makes each enclosed area four times as large.

Guide the blue tracer around a circle, an ellipse or a flower. Four hinged bars keep the coral pen on an enlarged copy of the same shape. Change the scale and compare distances, speeds and areas.

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

Rigid bars and frictionless pins impose exact planar geometry. The tracer follows a prescribed pattern; forces, inertia, joint play, bending and pen friction are omitted. Changing the scale rebuilds the bar lengths and pin positions.

Physics tutorial

## How four bars preserve a whole drawing

**Background** A pantograph copies a drawing with hinged rods. Its geometry connects every point of the original to a point in the copy.

**Why it matters** A machine can enlarge an entire curve without knowing whether it is tracing a circle or a flower.

Start with the essentials

- **Focus question**: Does doubling every distance also double the enclosed area?
- **One-sentence intuition**: The fixed pivot, tracer and pen stay collinear at a constant distance ratio. Lengths scale by that ratio and enclosed areas by its square.

### Core mathematical model

#### The parallelogram construction

$$
\mathbf D=\mathbf A+\mathbf v,\quad \mathbf B=k\mathbf A,\quad \mathbf C=\mathbf B+\mathbf v,\quad \mathbf Q=\mathbf B+k\mathbf v
$$

The fixed pivot is the origin. The two basic vectors have the short-link length; extending them by the same ratio makes the tracer and pen collinear with the origin.

#### Length, velocity and area scale

$$
\mathbf Q=k\mathbf D,\qquad \dot{\mathbf Q}=k\dot{\mathbf D},\qquad A_Q=k^2 A_D
$$

The area relation applies to the region inside a simple closed loop, or to each lobe separately. The scale is positive and fixed during each drawing.

### Common difficulties

#### A parallelogram needs correctly placed pivots

**Typical misconception** Any four bars in a parallelogram will copy at any chosen scale.

**Better mental model** The two long bars, their intermediate pin locations and the two shorter bars must be matched. The scale control changes this geometry together.

#### Area grows faster than length

**Typical misconception** A double-size circle has double the area.

**Better mental model** Both horizontal and vertical distances double. Every small patch becomes four times as large, so a simple enclosed area quadruples.

### Run the experiment

1. 01

   #### Trace a circle

   Choose Circle and move the blue tracer around its path.

   **What to observe:** The coral pen follows a larger circle. The three points stay on the same ray from the fixed pivot.

2. 02

   #### Change the whole shape

   Try Ellipse, Figure eight and Five-lobe flower without changing the scale.

   **What to observe:** The same bar arrangement copies every bend. Changing the path does not change the length or speed ratio.

3. 03

   #### Compare length and area

   Select Circle and change Drawing scale from two to three.

   **What to observe:** The copied radius becomes three times the original and the area nine times as large. The pen moves three times as fast as the tracer.

Physical conclusion

Similarity preserves angles and shape, but lengths, speeds and areas grow by different powers of the scale.

Sources: [US Army Engineer School · Engineering Drawing II, §6](https://files.eric.ed.gov/fulltext/ED221758.pdf) · [He et al. · Scissor lift virtual work](https://ietresearch.onlinelibrary.wiley.com/doi/full/10.1049/joe.2018.8961) · [MIT · Mechanical advantage](https://ocw.mit.edu/courses/2-007-design-and-manufacturing-i-spring-2009/29b95c6d8570f9aeb140c151747c96e3_MIT2_007s09_lec01.pdf)

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an 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)