Rolling geometry · Hypotrochoids

Tusi Couple & Inner Rolling Tracer

Can two circles draw an exact straight line? Yes. Roll a circle without slipping inside a fixed circle twice its radius, and a pen on its rim slides back and forth along a diameter of the fixed circle. The disk spins backward exactly as fast as its center orbits, so the vertical parts of the two motions cancel and the horizontal parts add. Move the pen inward and the line opens into an ellipse.

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.

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

Find the straight line hidden in two circles.

Orbit phase

\theta

235.00\,{}^\circ

Prescribed orbit rate

\omega

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

Fixed / rolling radius

n=R/r

2.00\,

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 inside makes the disk spin opposite to its orbit. A fixed circle twice the rolling radius, with the pen on the rim, draws the exact Tusi line. Its ends are points where the pen turns back, not ordinary sharp cusps. Moving the pen inward gives an ellipse; larger whole-number ratios give hypocycloids with several cusps.

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; the model does not work out what keeps it rolling. An ideal optical stylus projects the pen position onto a raised transparent sheet. Forces, friction, the motor, gear teeth, elasticity and loss of contact are not modeled.

Start with the essentials

Focus question: Can two circular motions cancel one coordinate exactly?
One-sentence intuition: When the fixed circle has twice the radius of the rolling one, the orbit and the spin turn at the same rate in opposite directions. For a pen on the rim, the vertical parts of the two motions cancel and the horizontal parts 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 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.

Tusi's exact cancellation

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

When the pen sits on the rim, the vertical coordinate stays zero. The line reaches one fixed-circle radius on each side of the origin.

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
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.
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