---
title: "Outer Rolling Tracer Interactive Lab | Sandbox Physics"
description: "One rolling circle. A whole family of flowers. Why does the pen stop at a cusp while its center moves?"
canonical_url: "https://sandboxphysics.com/labs/outer-rolling-tracer/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/outer-rolling-tracer/"
last_modified: "2026-10-22"
---

Rolling geometry · Epitrochoids

# Outer Rolling Tracer

Guide a disk around the outside of a fixed circle. Draw a cardioid, a nephroid or multi-cusp curves, then extend the pen past the rim to create loops. At a cusp the pen pauses while the center keeps moving.

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

Prescribed ideal planar rolling without slipping, with integer radius ratios. The elevated optical stylus may extend beyond the rolling rim. This is a geometric tracer, not a loaded gear or rotary engine simulation.

Physics tutorial

## Why does the pen stop at a cusp?

**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 the center move while a point on the rim is at rest?
- **One-sentence intuition**: The translating center and spinning radius contribute opposite velocities at contact. A rim pen passes through that instant of rest at each cusp.

### Core mathematical model

#### Circle center and spin

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

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.

#### Contact at a cusp

$$
d=r,\quad\theta=\frac{2\pi j}{n}\;\Longrightarrow\;\mathbf P=\mathbf K,\quad\dot{\mathbf P}=\mathbf 0
$$

Integer cusp phases bring a rim pen to instantaneous contact and rest.

#### 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

   #### Find the cusp

   Select cardioid and put the orbit phase at zero. Move the pen inward, then beyond the rim.

   **What to observe:** The rim pen stops at the cusp. Inset motion is smooth; the overhanging pen makes a loop.

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)