---
title: "Tusi Couple & Inner Rolling Tracer Interactive Lab | Sandbox Physics"
description: "Two circles can draw a straight line. Can two circles draw an exact straight line?"
canonical_url: "https://sandboxphysics.com/labs/inner-rolling-tracer/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/inner-rolling-tracer/"
last_modified: "2026-10-22"
---

Rolling geometry · Hypotrochoids

# Tusi Couple & Inner Rolling Tracer

Roll a circle inside a fixed ring. Move the pen, change the radius ratio, and watch an exact straight line unfold into ellipses, pointed stars and looping curves.

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

Prescribed ideal planar rolling without slipping. An ideal optical stylus projects onto an elevated transparent sheet. No force, friction, motor, tooth engagement, elasticity or contact-loss model.

Physics tutorial

## Where did the vertical motion go?

**Background** A point attached to a rolling circle combines the center's translation with the circle's own spin.

**Why it matters** Connect the contact condition to a visible curve, rather than treating the flower as a decorative formula.

Start with the essentials

- **Focus question**: Can two circular motions cancel one coordinate exactly?
- **One-sentence intuition**: With a two-to-one radius ratio, the two angular rates are equal and opposite. A rim pen's vertical components cancel, while its horizontal components add.

### Core mathematical model

#### Circle center and spin

$$
\mathbf C=(R-r)(\cos\theta,\sin\theta),\quad\phi=(1-n)\theta
$$

The radius ratio is a positive integer; internal rolling requires at least two. Spin is measured in the fixed frame.

#### The solved pen locus

$$
\mathbf P=\mathbf C+d(\cos\phi,\sin\phi),\quad d=kr,\quad n=R/r
$$

The pen has a fixed offset on the rolling body. Its initial radius points toward the contact on either side of the fixed circle.

#### Tusi's exact cancellation

$$
n=2,\quad\mathbf P=((r+d)\cos\theta,(r-d)\sin\theta)
$$

When the offset equals the rolling radius, the vertical coordinate vanishes. The segment extends one fixed-circle radius on each side of the origin.

#### No-slip material velocity

$$
\mathbf v_K=\dot{\mathbf C}+\dot\phi\,\hat{\mathbf z}\times(\mathbf K-\mathbf C)=\mathbf0,\quad\mathbf K=R(\cos\theta,\sin\theta)
$$

The contact marker traces a moving locus, but the material point there is instantaneously at rest against the fixed circle.

#### Rate scaling

$$
\mathbf v=\omega\frac{d\mathbf P}{d\theta},\quad\mathbf a=\omega^2\frac{d^2\mathbf P}{d\theta^2},\quad T=\frac{2\pi}{\omega}
$$

Analytic derivatives generate readouts, scene timing, plots and export from the same kinematics.

### Common difficulties

#### Contact location is not a material label

**Typical misconception** The moving contact marker must have the same velocity as the rolling surface material.

**Better mental model** Successive material points arrive at the marked location. Differentiate a rigid body's material point velocity at the instantaneous contact, not the sequence of contact positions.

#### No-slip is an assumption here

**Typical misconception** The picture proves friction can maintain contact at any speed.

**Better mental model** The center orbit and no-slip spin are prescribed. Required contact forces, friction limits, power, inertia and loss of contact need a separate dynamical model.

### Run the experiment

1. 01

   #### Follow the contact

   Choose contact view. Advance orbit phase slowly and inspect the velocity budget.

   **What to observe:** The location of contact advances, but the rolling material point at that instant has zero speed.

2. 02

   #### Open the line

   Select Tusi straight line, then move the pen inward without changing the radius ratio.

   **What to observe:** The cancelled vertical motion returns. The straight line becomes an ellipse.

3. 03

   #### Change shape, then scale time

   Try several integer radius ratios. Double the prescribed orbit rate and compare the path with velocity and acceleration.

   **What to observe:** The path stays fixed when only the rate changes. Velocity doubles and acceleration quadruples; the configuration still closes after one orbit.

Physical conclusion

The pen offset controls cusps and loops, while no-slip contact fixes the spin rate.

Geometry: [Robert Ferréol · Hypotrochoid](https://mathcurve.com/courbes2d.gb/hypotrochoid/hypotrochoid.shtml) · [Epitrochoid](https://mathcurve.com/courbes2d.gb/epitrochoid/epitrochoid.shtml)

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)