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.

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

Motion lies in the right/up plane; depth points toward the viewer. Positive orbit phase is counterclockwise from the right. Controls use degrees, while calculations and CSV use radians, metres and seconds. Wheel rotation is relative to the fixed laboratory frame.

Blue is the rolling circle, gray is the fixed circle, gold is a rigid pen arm, coral marks the pen and ink, and mint marks contact. The rear carrier prescribes the center orbit. The stylus and shaft stay below an elevated transparent sheet. A short ideal normal projection puts the trace directly above the solved pen position; overhanging arms clear the fixed circle.

Internal rolling spins opposite to the orbit. A fixed circle of twice the rolling radius, with the pen on the rim, produces the exact Tusi line. Its endpoints are reversals rather than ordinary sharp cusps. Moving the pen inward makes an ellipse; larger integer radius ratios make multi-cusp hypocycloids.

The contact location moves around the fixed circle, but the rolling body's material point instantaneously there has zero velocity. Its center translation and spin velocity cancel. The contact plot uses the positive orbit tangent; it is a velocity budget, not a force diagram.

Integer radius ratios make the full configuration repeat after one center orbit, though degenerate pen paths can repeat earlier. Playback stops after one interval. The faint full path is an ideal reference; bright ink is the same solved pen locus through the current phase. Curves and CSV are sampled analytic kinematics, not measured data.

No-slip rolling and the center orbit are imposed constraints. The model does not calculate the friction or normal force needed to maintain them. Circle radii and pen offsets are metrically consistent; thicknesses, ink width and hardware clearances are schematic. No tooth, force, inertia, gravity, slip or contact-loss dynamics are simulated.

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.

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 misconceptionThe moving contact marker must have the same velocity as the rolling surface material.

Better mental modelSuccessive 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 misconceptionThe picture proves friction can maintain contact at any speed.

Better mental modelThe 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

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.
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 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.

Geometry: Robert Ferréol · Hypotrochoid · Epitrochoid