Polar Equation Plotter

The Polar Equation Plotter graphs any expression of the form \( r = f(\theta) \) directly in your browser. Use it to draw the classic rosette \( r = \sin(3\theta) \), the heart-shaped cardioid \( r = 1 + \cos\theta \), Archimedean and Fermat spirals, limaçons with inner loops, lemniscates, and even the famous butterfly curve. Type your own expression with full sin, cos, tan, exp, log, sqrt support and the constants \( \pi \) and \( e \), or click one of nine presets for an instant plot. Overlay up to three equations on the same canvas, watch the live preview redraw as you type, then export the chart as crisp SVG or PNG.

How Polar Coordinates Work

A Gallery of Famous Polar Curves

What Makes This Polar Plotter Different

Expression Syntax — Quick Reference

Counting Petals on a Rose

For the rose curve \( r = \sin(k\theta) \) (or \( r = \cos(k\theta) \)) where \( k \) is an integer, the petal count follows a beautiful rule:

• If \( k \) is odd: the rose has exactly \( k \) petals.
• If \( k \) is even: the rose has \( 2k \) petals.

So \( \sin(3\theta) \) gives 3 petals, \( \sin(4\theta) \) gives 8 petals, and \( \sin(7\theta) \) gives 7. The reason is subtle: when k is odd, the petals drawn for negative r (which reflect through the origin) land back on the same positions as the positive-r petals. When k is even, the negative-r petals fill the gaps between the positive-r ones, doubling the count. Try \( \sin(2\theta) \) (4 petals) versus \( \sin(3\theta) \) (3 petals) to see the symmetry difference live.

From Cardioid to Limaçon: One Parameter Family

The general equation \( r = a + b\cos\theta \) traces a family of curves controlled by the ratio \( b/a \):

• \( b/a = 0 \): circle of radius \( a \) — no asymmetry.
• \( 0 < b/a < 1 \): dimpled limaçon — a slightly squashed oval.
• \( b/a = 1 \): cardioid — the perfect heart shape with a single cusp.
• \( 1 < b/a < 2 \): dimpled limaçon with a deeper indent.
• \( b/a \geq 2 \): limaçon with an inner loop — the curve crosses itself.

Try plotting \( r = 1 + b\cos\theta \) with b = 0.5, 1.0, 1.5, 2.0 in the three overlay slots to watch the heart bloom into a looped snail.

Real-World Uses

• Math classrooms: the animated draw reveal and live preview make polar equations physical — students see how the rotating radius traces out the curve.
• Physics labs: antenna radiation patterns, plant phyllotaxis, planetary orbits and pendulum traces all live in polar coordinates.
• Engineering: cam profiles, gear teeth and beam stress distributions are designed in polar form. Export SVG for laser cutting or CNC.
• Design and ornament: roses, lemniscates and butterfly curves make stunning logos, mandalas and pattern repeats. Export to vector for further editing.
• Generative art: overlay three rose curves at different k values in a neon palette for instant geometric posters.
• Astronomy: conic sections in polar form (\( r = p / (1 - e\cos\theta) \) for ellipse/parabola/hyperbola) describe planetary orbits — try it with eccentricity values from 0.1 to 0.9.

Tips for Beautiful Plots

• Pick the right θ range. Roses and cardioids close at 0 to 2π. Limaçons with inner loops can need 0 to 4π. Archimedean spirals look best at 0 to 8π or longer. Use the dropdown — it handles the multiples of π for you.
• Use overlay for "before/after" contrasts. Plot \( \sin(2\theta) \) and \( \sin(3\theta) \) side by side to see the even-vs-odd petal rule. Plot \( 1 + \cos\theta \) and \( 1 + 1.5\cos\theta \) to see a cardioid become a dimpled limaçon.
• Crank the resolution for spirals. Default Medium (1,800 samples) is plenty for roses. For long Archimedean or butterfly curves switch to High or Ultra — the extra samples reveal fine detail at the spiral edges.
• Lemniscates need both branches. Because the equation \( r^2 = 4\cos 2\theta \) has two square roots, plot \( \sqrt{4\cos(2\theta)} \) in equation 1 and \( -\sqrt{4\cos(2\theta)} \) in equation 2 to get both lobes.
• Hide the grid for portfolio art. Switch grid to "None" plus the Neon palette on a graphite background — the result feels like a generative-art print.

Frequently Asked Questions