SketchWave Limàçon Demo

Control Panel for SWLimacon Customization — Stage

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Control Panel

Limàçon Style & Attributes

(Drag the center dot to reposition the limàçon)

Offset (a): 5.00

0 (circle through origin) Large offset

Amplitude (b): 4.00

Small Large

Current variant: Dimpled

Ratio |a/b| = 1.25 → 1 < a/b < 2 = Dimpled

Stroke Thickness: 3

Stroke Color & Opacity

Opacity: 100

Fill Color & Opacity

Opacity: 55

Inner Loop Fill

Hide inner loop shading

Only applies to the Inner Loop variant. The outline (stroke) always shows both loops.

Equation Form

Cosine form: r = a + b·cos(θ) — toggle for sine

Sine form rotates the limàçon 90° CCW

Center Dot

Show center dot (pole)

Center = (0.0, 0.0)

Jump to a specific limaçon variant. The badge above updates live as you move the sliders.

Opacity (Alpha): 100

Transparent — enables motion trails

Speed (°/sec, −720 to 720): 30

CW (−) CCW (+)

Sinusoidally oscillates a around its slider value. At large depth values the curve morphs through variants (Inner Loop → Cardioid → Dimpled → Convex) in real time.

Speed (Hz): 0.8

Slow Fast

Depth (Δa): 2.00

Subtle Dramatic (variant morphing)

SWLimacon is a SketchWaveJS shape class representing the complete limaçon family of polar curves, defined by r = a + b·cos(θ) (or r = a + b·sin(θ)). The name comes from the Latin limax (snail), coined by Gilles de Roberval. The curve was first studied by Étienne Pascal, father of Blaise Pascal.

The Five Variants — determined by the ratio |a/b|

Condition	Variant	Example (a, b)	Description
`a/b > 2`	Convex	a=8, b=3	Oval-like; r always positive; smooth outer boundary, no indentation
`a/b = 2`	Trisectrix	a=6, b=3	Special convex limaçon historically used to trisect angles; has an inflection point on the right
`1 < a/b < 2`	Dimpled	a=7, b=5	r stays non-negative but dips inward, creating a visible concavity (dimple) on the left side
`a/b = 1`	Cardioid	a=5, b=5	r just touches zero — one cusp at the pole; the familiar heart-shaped curve; also modelled by SWCardioid
`a/b < 1`	Inner Loop	a=3, b=5	r goes negative for a range of θ, creating a second inner loop inside the outer curve

Negative r and the Inner Loop

When a < b, the minimum value of r is a − b < 0. In polar coordinates, a negative r places the point in the opposite direction from θ, which is exactly what creates the inner loop. SWLimacon computes x = r·cos(θ), y = r·sin(θ) directly — negative r is handled automatically with no special-case logic.

Controls

Offset (a): the additive constant. Increasing a moves toward convex; decreasing moves toward inner loop.
Amplitude (b): the oscillation amplitude. Increasing b deepens the variation; decreasing b flattens toward a circle.
Variant Badge: updates live as you move the a and b sliders — shows the current variant with a color-coded label.
Variant Presets: jump instantly to Convex, Trisectrix, Dimpled, Cardioid, or Inner Loop.
Equation Form: toggle between cosine (opens right) and sine (rotated 90° CCW, opens upward).

Animations

Spin: continuously rotates the limaçon about its center (pole). Open the Spin accordion to set speed (°/second).
Pulse: sinusoidally oscillates a around its slider value. At large depth values this causes the curve to morph through variants in real time — watch Inner Loop → Cardioid → Dimpled → Convex cycle continuously.
Motion trails: lower the background opacity to create ghostly trails during animation.

Interactive features

Drag the center dot (the pole) to reposition the entire limaçon.
Snap Center rounds the pole to the nearest grid intersection.
Click the canvas (not on the center dot) to toggle the grid.
Keyboard: s (spin toggle), p (pulse toggle), r (factory reset), q (stop all animations).

SketchWaveJS classes used by this demo

SketchWave Limàçon Demo

Control Panel

Limàçon Style & Attributes

Variant Presets

Background

Spin (Rotation)

Pulse (Variant Morphing)