Epicycloid Family

Six epicycloid shapes — from Nephroid to 7-Petal Rosette — Stage

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Color Scheme

Radius: 3.5 grid units

1.0 5.5

Speed: 30°/sec

CW fast CCW fast

Speed: 0.5 Hz

slow fast

Amount: 1.0 grid units

subtle dramatic

Background: 100% (low = ghosted trails)

trails solid

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What Is an Epicycloid?

An epicycloid is the curve traced by a point on the rim of a small circle of radius a as it rolls around the outside of a fixed circle of radius R. The number of cusps (petal tips) equals the ratio R/a.

The Parametric Equations

For a clover with k petals, SWClover uses:

n = k + 1          (internal epicycloid parameter)
a = radius / k     (rolling-circle radius, scaled so each tip = radius from centre)

x(t) = a · (n · cos(t) − cos(n · t))
y(t) = a · (n · sin(t) − sin(n · t))

t ∈ [0, 2π)

Increasing k by 1 adds one more petal — and requires the rolling circle to complete one more loop around the fixed circle.

The Six Forms in This Gallery

k (numPetals)	n (internal)	Shape name	Historical note
2	3	Nephroid — 2-cusped kidney shape	First described by Christiaan Huygens (1678)
3	4	3-Petal ♣ — club suit shape	—
4	5	Clover ☘ — the default SWClover	4-leaf clover; 4-cusp epicycloid
5	6	5-Petal — pentagonal rosette	—
6	7	6-Petal — hexagonal flower	—
7	8	7-Petal — dense rosette	Approaching a near-circle boundary

The Rolling Circle Model

Imagine a fixed circle of radius R and a small circle of radius a = R/k rolling around its outside edge. A pen attached to a point on the rim of the rolling circle traces the epicycloid. Each complete roll of the small circle produces one cusp (petal tip).

R/a = 1 → n = 2: cardioid (1 cusp) — not in this gallery, but the starting point of the family.
R/a = 2 → n = 3: nephroid (2 cusps) — top-left in the gallery.
R/a = 3 → n = 4: 3-petal (3 cusps).
R/a = 4 → n = 5: 4-leaf clover (4 cusps).
… and so on.

The term epicycloid was introduced by the Danish astronomer Ole Rømer around 1674 while he was studying gear tooth shapes. Epicycloids power the Spirograph toy and appear in the mathematics of complex analysis and gear design.

Animation Controls

Spin — rotates all six shapes simultaneously. Negative speed reverses direction.
Breathe — oscillates the petal radius sinusoidally, like a heartbeat. All six shapes share the same radius at any given moment, making the breathing synchronised.
Background alpha — reduce to create ghosted motion trails. Set to 0 and enable spin for a mandala-like effect.
Radius — sets the petal-tip distance for all six shapes.
Color Scheme — four presets: Forest (green), Ocean (blue), Sunset (red-orange), Amethyst (purple).

Keyboard Shortcuts

s — toggle spin
b — toggle breathe
r — factory reset
q — stop all animations

Related: SWClover Demo · SWClover Reference

Epicycloid Family

☰ Control Panel

Color Scheme

Spin

Breathe