Polar Graph Generator | r(θ)

Polar coordinates (r, θ) define a point by its distance r from the origin (called the pole) and its angle θ measured counterclockwise from the positive x-axis. Unlike Cartesian (x, y) coordinates that use perpendicular distances from two axes, polar coordinates describe position using rotation and distance, natural for circular and spiral geometry. Conversion to Cartesian: x = r·cos(θ), y = r·sin(θ). Inverse: r = √(x²+y²), θ = atan2(y, x).

• ›A circle centred at the origin: simply r = constant in polar form (vs x² + y² = r² in Cartesian)
• ›A spiral: r = aθ, radius grows linearly with angle, impossible to write elegantly in Cartesian
• ›A cardioid: r = 1 + cos(θ), just 12 characters vs a complex implicit Cartesian equation
• ›Angle θ is measured in radians: 0 = east (right), π/2 = north (up), π = west, 3π/2 = south
• ›Negative r values plot the point in the opposite direction: r = −1 at θ = 0 plots at (1, π)

Use t as the angle variable θ. The expression is evaluated as a JavaScript expression, so all standard JavaScript Math functions are available, either directly (e.g. sin(t)) or prefixed with Math (e.g. Math.sin(t)). Arithmetic operators +, −, *, /, and ** (power) are all supported, along with parentheses for grouping.

• ›Trig: sin(t), cos(t), tan(t)
• ›Roots and powers: sqrt(t), pow(t, 2), t**3
• ›Other: abs(t), exp(t), log(t) (natural log)
• ›Constants: Math.PI ≈ 3.14159, Math.E ≈ 2.71828
• ›Example: 2*cos(3*t) + sin(t), exp(cos(t)) - 2*cos(4*t)

For r = cos(nθ), the petal count depends on whether n is odd or even. If n is odd, the curve has exactly n petals drawn over θ ∈ [0, π], because the negative r values from π to 2π exactly retrace the same petals already drawn in the first half. If n is even, the curve has 2n petals and requires θ ∈ [0, 2π] to draw all of them, because the negative-r values trace a new set of petals rather than retracing the existing ones.

• ›r = cos(3t), θ from 0 to π: 3 petals, correct (going to 2π traces each petal twice)
• ›r = cos(4t), θ from 0 to 2π: 8 petals (2×4), needs the full revolution
• ›r = cos(5t), θ from 0 to π: 5 petals; to 2π: each petal drawn twice (still looks like 5)
• ›r = sin(nθ) follows the same rules but rotated 90°/n relative to the cos version

A cardioid is the curve r = a(1 + cos(θ)), a heart-shaped closed curve generated geometrically by tracing a point on a circle of radius a as it rolls around another circle of the same radius. The name comes from the Greek "kardia" (heart). It is a special case of the limaçon r = a + b·cos(θ) when a = b. The cardioid has one cusp (the pointed bottom) where the rolling circle passes through the point of tangency.

• ›Mandelbrot set: the boundary of the main bulb is exactly a cardioid in the complex plane
• ›Microphone directivity: cardioid microphones pick up sound from the front and reject the rear
• ›Coffee cup caustic: the bright crescent formed by reflections of light on the inside of a mug
• ›Envelope of circles: the cardioid is the envelope of all circles passing through the origin whose centres lie on a fixed circle

The Archimedean spiral r = aθ grows linearly with angle, constant spacing between successive turns. The Fermat spiral r = √θ grows as the square root, so turns get progressively closer together as r increases. The logarithmic (equiangular) spiral r = e^(bθ) keeps a constant angle between the tangent and the radial line, seen in nautilus shells and galaxy arms. Use larger θ ranges to show more turns of any spiral.

• ›Archimedean: enter t/(2*Math.PI) with θ from 0 to 12.57 (2 turns), constant turn spacing
• ›Fermat: enter sqrt(t) with θ from 0 to 12.57, inner turns are wider, outer turns compress
• ›Logarithmic: enter exp(0.2*t) with θ from 0 to 12.57, self-similar spiral, same shape at all scales
• ›Hyperbolic spiral r = a/θ: starts far out and converges toward the origin as θ increases

The butterfly curve r = e^cos(θ) − 2cos(4θ) + sin⁵(θ/12) contains the term sin(θ/12). The period of sin(θ/12) is 2π × 12 = 24π, the curve only returns to its starting point after 24π radians of θ. Using θ from 0 to 12π (≈ 37.7) captures about half the complete curve, enough to see most of the wing structure. The full curve requires θ from 0 to 24π ≈ 75.4. Using shorter ranges shows increasingly incomplete outlines of the wings.

• ›θ from 0 to 2π: one-twelfth of the full curve, just a small segment
• ›θ from 0 to 12π: roughly half the curve, most of the wing structure visible
• ›θ from 0 to 24π: the complete curve closes back on itself
• ›The sin⁵(θ/12) term is responsible for the slow variation giving the wing texture
• ›Increasing sample count to 2000–3000 gives smoother rendering for this complex curve

This calculator plots one curve at a time on a fresh canvas. To visually compare two curves side by side, plot the first, download it as a PNG, then plot the second. You can overlay the two images in any image editor. For interactive multi-curve polar plotting, where both curves update together as you change parameters, dedicated graphing tools offer that capability while this calculator focuses on making each individual curve as detailed and downloadable as possible.

• ›Download PNG after each plot to preserve curves for comparison
• ›Desmos (desmos.com/calculator): free online graphing with full polar multi-curve support
• ›GeoGebra: supports polar curves and geometric constructions simultaneously
• ›Wolfram Alpha: type "polar plot r=1+cos(t), r=2cos(t)" for instant side-by-side comparison