SketchWave Bumpy Circle Demo

Control Panel for SWBumpyCircle Customization — Stage

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Control Panel

BumpyCircle Parameters

(Drag the center dot to reposition the figure)

Radius (base circle): 5.0

Small Large

Amplitude (bump height): 1.5

None (0) Large (4)

Frequency (bumps/revolution): 6

1 bump 20 bumps

Power (bump shape): 1

Sinusoidal (1) Pointed (9)

Rotation (°): 0

0° 360°

Stroke Thickness: 2

Stroke Color & Opacity

Opacity: 100

Fill Color & Opacity

Opacity: 40

Overlays

Show center dot

Show base circle (dashed)

Center = (0.0, 0.0)

Opacity (Alpha): 100

Transparent — enables motion trails

Speed (°/sec, −720 to 720): 45

CW (−) CCW (+)

Sinusoidally oscillates the base radius and/or bump amplitude over time. Try combining Breathe with Spin and a low Background opacity for flowing wave trails!

What breathes?

Radius Amplitude Both

Radius Settings

Speed (Hz): 0.5

Slow (0.1 Hz) Fast (2.0 Hz)

Amount (grid units): 1.5

Subtle Dramatic

Amplitude Settings

Speed (Hz): 0.5

Slow (0.1 Hz) Fast (2.0 Hz)

Amount (grid units): 1.0

Subtle Dramatic

What is a Bumpy Circle?

A bumpy circle starts with a plain circle of radius r and adds a sinusoidal ripple to that radius as a function of angle θ. The result is a smooth, closed wave-like silhouette with a controllable number of bumps, bump height, and bump sharpness. It is one of the simplest examples of a polar curve — where the distance from the center varies with direction.

The Polar Equation

r(θ) = radius + amplitude · sin(frequency · θ)^power
x(θ) = r(θ) · cos(θ)
y(θ) = r(θ) · sin(θ)
θ ∈ [0, 2π)

At each angle θ, the radius is the base circle radius plus a sinusoidal perturbation. The four shape parameters each have a distinct visual effect:

Parameter	Effect	Default
radius	Base circle size (grid units)	5
amplitude	Peak height of each bump above the base circle; 0 = plain circle	1.5
frequency	Number of bumps per full revolution; integer values give perfect closure in one turn	6
power	Exponent applied to sin(…): 1 = smooth bumps & dents, even = bumps only (all outward), odd > 1 = pointed bumps with gentler dents	1

Why Does the Curve Close?

For the shape to meet itself perfectly after one revolution, the sine term must complete a whole number of cycles in [0, 2π). With frequency = n (a positive integer), sin(nθ) completes exactly n full cycles as θ goes from 0 to 2π, producing n bumps and returning the radius to its starting value. Integer frequency values always give a perfectly closed curve in one revolution.

Effect of the Power Parameter

The power exponent changes the shape of each individual bump:

power = 1: Standard sinusoid — peaks above the base circle alternate with troughs (dents) below. The most wave-like appearance.
power = 2: sin² is always non-negative, so all perturbations are outward bumps. The circle never dips inward. Bumps are slightly flatter and wider than power = 1.
power = 3: More pointed bumps, with shallower but still-present dents.
power = 4+: Bumps become increasingly narrow and spike-like, with the base circle growing more prominent between them.

Polar Curves in Mathematics

A polar curve r(θ) expresses distance from the origin as a function of angle. The bumpy circle is closely related to:

Rose curves: r = cos(nθ) — petals from zero on the sides.
Limacon: r = a + b·cos(θ) — an offset circle with a single smooth bulge or dimple.
Rhodonea: generalized rose-like forms with varying petal widths.

SWBumpyCircle is like a “limaçon with multiple lobes” — a circle base with sinusoidal modulation at an arbitrary harmonic.

Using the Controls

Radius — scales the overall circle. Breathing oscillates this value.
Amplitude — how far the bumps protrude above the base circle. Set to 0 for a plain circle.
Frequency — number of bumps around the circle. Use integer values (1–20) for perfect closure. Try frequency = 1 for a limaçon-like shape.
Power — bump sharpness. 1 = smooth waves with dents, 2 = all-outward bumps, higher = spikier.
Rotation — rotates the entire bump pattern CCW. With odd frequency, notice that bumps and gaps trade places at 180°.
Show base circle — overlays a dashed circle at the base radius, making it easy to see exactly how far each bump protrudes above (or dips below) the unperturbed circle.
Breathe (accordion) — sinusoidally oscillates the base radius, the bump amplitude, or both over time. Each has its own Speed and Amount sliders. Try combining with Spin and a low Background opacity for flowing spiral trails!
Spin (accordion) — continuously rotates the bump pattern.
Drag the center dot to reposition the figure.
Click the canvas to toggle the grid on/off.

Keyboard Shortcuts

s — Toggle spin
b — Toggle breathe
r — Factory reset
q — Stop all animations