You sketch a curve from polar equations the way a draftsman would: pick a family, twist the knobs, watch the line breathe. Nothing uploads. The canvas stays in your tab so you learn by dragging, not by reading a wall of formulas first.
Archimedean: Radius grows at a steady rate as you sweep angle. Good first curve when teaching polar plots because the algebra stays linear.
Shown equation: r = a + bθ
A Fibonacci spiral you see in design blogs is usually a polyline made of quarter circles, while the golden option here stays a smooth exponential sweep. Both borrow φ from the same number family you get in the golden ratio calculator, yet they are not interchangeable when someone asks for “the” golden spiral in a strict geometry class.
The hyperbolic branch needs a cutoff near θ → 0 because the ideal radius shoots toward infinity. Your preview clamps and rescales so the browser never divides by a true zero. Treat exported PNGs as illustrations, not lab measurements.
Art students testing tattoo or logo rhythm often want a quick polar doodle without installing MATLAB. Teachers demoing polar coordinates pair this page with hand calculations so learners see the same θ stepping forward.
Developers prototyping procedural textures sometimes snapshot a rainbow-mode spiral, then feed the bitmap into a shader as a mask. The workflow sounds niche until you count how often UI backgrounds borrow slow curls.
Each frame samples hundreds of polar steps, converts (r,θ) to pixels, joins them with short segments, and optionally restrokes per segment for rainbow modes. Length sums those segment sizes; “shoelace area” closes the polyline against the origin for a rough fill estimate. Neither value claims calculus-grade precision, especially when line thickness overlaps itself.
Animations temporarily reduce the turn count, ease the curve forward, then restore your slider. If you spam the button, the guard waits for the prior run to finish so frames stay smooth on phones.
| Family | Shape of rule | When you might choose it |
|---|---|---|
| Archimedean | r = a + bθ | Even spacing between arms; feels like a coiled rope. |
| Logarithmic | r = ae^(bθ) | Self-similar growth; echoes shells and galaxies in stylized art. |
| Fibonacci style | φ-based exponential | Quick stand-in when you want golden growth without drawing squares. |
| Golden | Logarithmic with φ-tied growth | Smoother cousin to the rectangle spiral; pairs with Fibonacci number play. |
| Hyperbolic | r ∝ 1/θ | Dramatic collapse toward the pole; needs the built-in floor. |
| Fermat | r² ∝ θ | Parabolic spiral; satisfying S-curve when you push turns high. |
We do not verify physical models, export vector files, or solve for parameters from a photo you traced. If you need tiling patterns with symmetry groups, jump to the geometric pattern generator instead; spirals only cover a slice of decorative math.
Ask students to lock a = 1, sweep b from 0.05 to 0.4 on an Archimedean spiral, and describe how many times the curve hits the edge of the square viewport before three full turns finish. Then switch to logarithmic with the same turn count, bump b by 0.02 steps, and compare how fast the arms appear to “tighten.” The numbers stay humble, yet the contrast between linear and exponential radius rules shows up immediately.
For a follow-up, have them export two PNGs at 1280 px wide screens and overlay them in an editor with multiply blend. You get a moiré lecture without buying plotting software seats.
Short answers about accuracy, files, and classroom use.
The ideal radius grows without bound as the angle approaches zero, so the preview clamps with a small positive angle floor. You still see the characteristic hook; you do not get a singularity on screen.
No. Length adds pixel segment lengths along the sampled polyline. Area uses a shoelace sum with the origin, so thick strokes, self-crossings, or extreme parameters skew both numbers. Use them for intuition, not certification.
PNG raster at the current canvas resolution, respecting your device pixel ratio up to a cap of two for sharper phones. There is no SVG or PDF path yet.
The animation temporarily drives the turn slider, then restores your previous value. If you edit sliders mid-animation, wait until the motion stops so values snap back predictably.
Neither option draws the classic square tiling with quarter-circle arcs. Fibonacci style uses a smooth φ-powered radius sweep; golden ties growth rate directly to φ per radian step. For arc-based diagrams you still need compass construction or CAD.
No signup runs here. Load the page, move sliders, export if you want. Offline use depends on your browser cache for assets, but the math itself stays client-side once scripts load.
