Polar Curves

Rose, cardioid, spiral — type r = f(t) to explore

Polar curves describe shapes using distance from the origin r as a function of angle θ. Instead of y = f(x), you write r = f(θ) — and stunning curves emerge.

This gallery starts with six classic curves: the cardioid (heart), rose curves (petals), the lemniscate (infinity), the limaçon (snail with inner loop), and the logarithmic spiral (nature's favorite curve).

Type your own polar equations in the function input: r = cos(2*t) for a 4-petal rose, r = t for an Archimedean spiral. Use t for θ. The AI assistant can explain why odd and even petals differ, or what makes a spiral equiangular.

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What is a polar curve?
A polar curve is a graph defined by r = f(θ), where r is the distance from the origin and θ is the angle. Unlike Cartesian graphs (y vs x), polar curves can create loops, petals, and spirals naturally.
Why does r = cos(2θ) have 4 petals but r = sin(3θ) has 3?
For rose curves r = cos(nθ) or r = sin(nθ): if n is even, the curve has 2n petals (cos(2θ) → 4 petals). If n is odd, it has n petals (sin(3θ) → 3 petals). This is because even n traces petals in all four quadrants, while odd n overlaps half of them.
What is a lemniscate?
The lemniscate of Bernoulli has equation r² = a²cos(2θ). It forms a figure-eight (∞) shape. The curve only exists where cos(2θ) ≥ 0, creating two symmetric loops. It was studied by Jakob Bernoulli in 1694.
Where do logarithmic spirals appear in nature?
The logarithmic spiral r = e^(bθ) appears in nautilus shells, sunflower seed arrangements, hurricane patterns, and galaxy arms. It's also called an equiangular spiral because it crosses every radial line at the same angle — a property that makes it self-similar at every scale.
What is a limaçon?
A limaçon (French for "snail") has the form r = a + b·cos(θ). When |b| > |a|, an inner loop appears — the curve crosses the origin and loops back. When |b| = |a|, you get a cardioid. When |b| < |a|, it's a dimpled or convex curve.
Where are polar curves used in real life?
Polar curves appear everywhere: antenna radiation patterns (signal strength vs direction), radar displays, nautilus shells and galaxy arms (logarithmic spirals), mechanical cam profiles (cardioid cams), wind roses in meteorology, and the Fibonacci/golden spiral in sunflowers, pinecones, and hurricanes.
How do I convert a polar equation to cartesian?
Use the substitutions: x = r·cos(θ), y = r·sin(θ), and r² = x² + y². For example, the circle r = 2cos(θ) becomes r² = 2r·cos(θ), which is x² + y² = 2x, or (x−1)² + y² = 1 — a circle centered at (1, 0) with radius 1.
What can it graph?
It can plot explicit, implicit, and parametric functions, add points and geometry, and animate sliders on the same graph.
Can I use voice or a photo?
Yes. You can talk to the tutor, upload a worksheet or handwritten problem, and let the graph update from that input.
Will it explain the steps?
Yes. The AI explains what it is drawing and why, so you see the answer on the graph instead of getting only a final number.