Aesthetic Equations

About These Curves

• Love Equation — a heart traced from a single sinusoidal parametrization; the classic.
• Butterfly Curve — discovered by Temple Fay, it needs a full $24\pi$ sweep to complete its wings.
• Rose Curve — the polar equation $r=\cos(5\theta)$ draws a 5-petaled flower.
• Astroid — the path traced by a point on a small circle rolling inside a circle 4× its size.
• Lissajous Figure — two perpendicular sine waves at a 3:2 frequency ratio, the pattern behind oscilloscope art.
• Deltoid — a hypocycloid with 3 cusps, traced by a circle rolling inside one 3× its size.
• Cardioid — the polar equation $r=2(1+\cos\theta)$ traces the classic single-lobed "heart" curve, the epicycloid with one cusp.
• Nephroid — a 2-cusped epicycloid, literally Greek for "kidney-shaped".
• Spirograph Flower — a hypotrochoid with a pen offset larger than the rolling circle, looping outward into a 5-petaled flower over 3 full turns.
• Archimedean Spiral — radius grows linearly with angle ($r=0.3\theta$), the only open (non-closed) curve here.
• Drag the Points slider to trade off smoothness vs. performance, and switch Drawing Mode to see the individual sampled points instead of connecting lines.