Rolling geometry · Epitrochoids

Outer Rolling Tracer

Why does the pen stop at a cusp while its center moves? Because at a cusp the pen sits exactly on the contact point, and a disk rolling without slipping has zero speed there. The center moves forward while the spin carries that rim point backward, and the two cancel for an instant. Move the pen inside the rim and the cusps round off; push it past the rim and they turn into loops.

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.

Pen horizontal position $0$

Pen vertical position $0$

Pen speed $0$

Pen acceleration magnitude $0$

Signed spin / orbit rate $0$

Material contact-point speed $0$

One orbit duration $0$

Geometric constraint residual $0$

Roll & draw

Move the pen to change the flower.

Orbit phase

\theta

255.00\,{}^\circ

Prescribed orbit rate

\omega

45.00\,{}^\circ/\mathrm s

Fixed / rolling radius

n=R/r

3.00\,

Pen offset / rolling radius

k=d/r

1.00\,

Fixed-circle radius

R

1.20\,\mathrm m

The motion lies in a plane where right and up are positive, and depth points toward you. Orbit phase is measured counterclockwise from the right. The controls use degrees; the calculations and CSV use radians, metres and seconds. Wheel rotation is measured against the fixed laboratory frame.

Blue is the rolling circle, gray the fixed circle and gold the rigid pen arm; coral marks the pen and its ink, and mint marks the contact. The carrier at the back sets the center's orbit. The stylus and shafts stay below a raised transparent sheet. A short, ideal perpendicular projection puts the trace directly above the computed pen position, and arms that reach past the rim pass over the fixed circle.

Rolling outside makes the disk spin the same way it orbits. Equal radii with the pen on the rim draw a cardioid; a radius ratio of two draws a nephroid. Other whole-number ratios give the matching number of cusps. Moving the pen inward rounds off the cusps, and moving it past the rim makes loops.

The contact point moves around the fixed circle, but the point of the rolling disk that is there at each instant has zero velocity: the velocity from the center's motion and the velocity from the spin cancel. The contact plot takes the forward orbit direction as positive. It breaks down velocities and is not a force diagram.

With a whole-number radius ratio, the whole machine returns to its starting state after one orbit of the center, though some special pen paths repeat sooner. Playback stops after one interval. The faint full path is an ideal reference; the bright ink is the same computed path up to the current phase. The plots and CSV are samples of exact kinematics, not measured data.

Rolling without slipping and the center's orbit are imposed. The model does not compute the friction or normal force needed to keep them. Circle radii and pen offsets are drawn to scale; thicknesses, ink width and hardware clearances are schematic. Gear teeth, forces, inertia, gravity, slipping and loss of contact are not simulated.

The disk is made to roll in a plane without slipping, and the radius ratio is always a whole number. The raised optical stylus can reach past the rim of the rolling disk. This is a machine for drawing geometric curves, not a simulation of loaded gears or a rotary engine.

Start with the essentials

Focus question: Can the center move while a point on the rim is at rest?
One-sentence intuition: At the contact point, the velocity from the center's motion and the velocity from the spin cancel. A pen on the rim passes through that moment of rest at every 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 whole number, at least two for rolling inside. Spin is measured against the fixed frame.

Equation of the pen's path

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

The pen sits at a fixed distance from the center of the rolling disk. At the start, its radius points toward the contact point, whether the disk rolls inside or outside.

Contact at a cusp

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

At these phases a pen on the rim reaches the contact point and stops for an instant.

Velocity of a material point without slipping

\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 moves around the fixed circle, but the material point there is at rest against it for that instant.

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}

The readouts, the animation timing, the plots and the export all come from exact derivatives of the same motion.

Common difficulties

The contact point is not one fixed piece of the rim

Typical misconceptionThe moving contact marker must move as fast as the material of the rolling rim.

Better mental modelDifferent points of the rim arrive at the marker one after another. The material's velocity is the velocity of the point of the disk touching right now, not the rate at which the contact position moves.

No-slip is an assumption here

Typical misconceptionThe picture proves friction can maintain contact at any speed.

Better mental modelHere the center's orbit and the no-slip spin are simply imposed. The contact forces they require, the friction limit, power, inertia and loss of contact would need a separate dynamical model.

Run the experiment

01
Follow the contact
Choose the contact view. Advance the orbit phase slowly and watch the velocity breakdown.
What to observe: The contact point keeps moving forward, yet the point of the disk touching the fixed circle has zero speed at that instant.
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.
03
Change the shape, then the speed
Try several whole-number radius ratios. Then double the orbit rate and compare the path, the velocity and the acceleration.
What to observe: Changing only the rate leaves the path unchanged. The velocity doubles, the acceleration grows four times, and the machine still returns to its starting state after one orbit.

Geometry: Robert Ferréol · Hypotrochoid · Epitrochoid