Rolling-Circle Epicycloid

Watch a point trace its epicycloid path — one rolling revolution at a time — Stage

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Color Scheme

Petals: k = 3 (3-Petal, r = 1.17)

2 (Nephroid) 7-Petal

Fixed Circle Radius R: 3.5 units

2.0 4.5

Speed: 60 °/sec

15°/s 360°/s

Reverse direction (CW)

Trace Weight: 2.0 px

Fixed circle

Rolling circle

Spoke (arm to trace point)

Tracing dot

Trace path

Orbit ring (rolling-centre path)

Grid & axes

The Rolling-Circle Construction

An epicycloid is the curve traced by a point on the rim of a small circle rolling around the outside of a fixed circle. This page shows that construction live:

The fixed circle (radius R) sits at the centre.
The rolling circle (radius r = R/k) orbits without slipping.
The bright dot marks the tracing point on the rolling circle's rim.
The spoke line connects the rolling-circle centre to the dot, so you can watch it spin.
The orbit ring (dashed) shows the circular path of the rolling-circle centre.

Parametric Equations

With revolution angle t ∈ [0, 2π), rolling-circle radius r = R/k, and n = k + 1:

Rolling-circle centre:
Cx(t) = (R + r) · cos(t)
Cy(t) = (R + r) · sin(t)

Tracing-point position:
Px(t) = (R + r) · cos(t) − r · cos(n · t)
Py(t) = (R + r) · sin(t) − r · sin(n · t)

This is identical to the SWClover formula a·(n·cos(t) − cos(n·t)) with a = R/k. The tracing point starts at (R, 0) — the initial contact point on the fixed circle — so each petal's cusp touches the fixed circle exactly.

Shape Table

k (petals)	r = R/k	n = k+1	Shape Name	Historical Note
2	R/2	3	Nephroid	Huygens (1678); Greek for "kidney-shaped"
3	R/3	4	3-Petal ♣	Club-suit silhouette
4	R/4	5	4-Leaf Clover ☘	Default SWClover
5	R/5	6	5-Petal Rosette
6	R/6	7	6-Petal Rosette
7	R/7	8	7-Petal Rosette	Dense mandala petal

Controls Quick Reference

Petals (k)	Changes the number of cusps and the rolling-circle radius.
Fixed Radius (R)	Scales both circles while keeping the k ratio constant.
Speed	Revolution speed in degrees/second. Use keyboard `+` / `−`.
Reverse	Rolls the circle clockwise instead of counter-clockwise.
Show/Hide	Toggle individual visual layers to focus on the geometry.
Clear	Wipes the accumulated trace to start fresh.
Space	Play / Pause the animation.

The term "epicycloid" was coined by Ole Rømer around 1674 while studying gear-tooth profiles. The nephroid (k = 2) was named and fully described by Christiaan Huygens in 1678.

Rolling-Circle Epicycloid

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Color Scheme

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