Polar Equation Plotter
Plot polar equations interactively — graph r = sin(3θ), r = θ (Archimedean spiral), cardioids, limaçons, lemniscates and butterfly curves with adjustable θ range, sampling resolution, color palettes and polar grid. Overlay up to three equations on the same canvas and export the chart as crisp SVG or PNG.
\( x = r \cos\theta, \quad y = r \sin\theta \)
The chart above was rendered by sampling each equation at 1800 evenly spaced θ values across θ ∈ [0 to 2π], then drawing one continuous SVG path per curve.
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About 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
Every point on the plane has two equivalent labels. Cartesian coordinates \( (x, y) \) say "go this far right and that far up". Polar coordinates \( (r, \theta) \) say "go this far out from the origin, at this angle from the positive x-axis". The two are linked by
\[ x = r\cos\theta, \quad y = r\sin\theta \]
A polar equation \( r = f(\theta) \) declares the radius as a function of the angle. The plotter sweeps θ across the chosen range, evaluates \( f \) at each step, converts the resulting \( (r, \theta) \) to \( (x, y) \), and connects the dots with a single SVG path. The animated dot above shows exactly that — the violet radius rotates with θ, and the pink dot at distance r leaves the trace.
A Gallery of Famous Polar Curves
What Makes This Polar Plotter Different
2cos(3t), theta^2, 1 + 2cos(θ). Implicit multiplication, caret-power and Unicode θ/π are all converted automatically — no syntax cheat-sheet needed.
Expression Syntax — Quick Reference
| What you type | Meaning | Example |
|---|---|---|
theta or t or θ | The polar angle (in radians) | r = theta |
pi or π | The constant π ≈ 3.14159 | r = sin(theta + pi/4) |
e | Euler's number ≈ 2.71828 | r = exp(theta/5) |
sin, cos, tan | Trig functions (radians) | r = sin(3*theta) |
asin, acos, atan, atan2 | Inverse trig | r = atan(theta) |
exp, log, log2, log10 | Exponential & logarithms | r = log(theta + 1) |
sqrt, abs, floor, ceil | Power & rounding | r = sqrt(abs(cos(2*theta))) |
^ or ** | Exponentiation | r = theta^2 |
Implicit * | Number-to-letter inserts × | 2cos(3t) → 2*cos(3*t) |
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
What is a polar equation?
A polar equation defines a curve as a relationship between the distance r from the origin and the angle θ (measured counter-clockwise from the positive x-axis). Examples: r = sin(3θ) traces a three-petal rose; r = 1 + cos(θ) draws the heart-shaped cardioid; r = θ spirals outward as the Archimedean spiral. Each point (r, θ) maps to Cartesian coordinates via x = r cos θ, y = r sin θ.
What functions can I use in the expression?
You can use sin, cos, tan, asin, acos, atan, atan2, sinh, cosh, tanh, exp, log, log2, log10, sqrt, abs, floor, ceil, pow, min and max — all the standard math functions. Constants pi, e and tau are available, plus the variable theta (you can also write t as a shortcut, and the Unicode θ symbol is converted automatically). All trig is in radians.
How do I write implicit multiplication?
The parser handles it automatically: 2cos(3t), 3theta, 2.5pi all work as expected — no need to type the * between a number and a letter or parenthesis. You can also use the caret ^ for powers, so theta^2 is the same as theta**2. This lets you copy equations from textbooks without rewriting them.
What is the petal count for r = sin(kθ)?
For r = sin(kθ) or r = cos(kθ) with integer k: if k is odd, the rose has exactly k petals; if k is even, it has 2k petals. So sin(3θ) gives 3 petals, sin(4θ) gives 8 petals, and sin(7θ) gives 7. This is because negative r reflects through the origin — odd k retraces the same petals while even k draws new ones in between.
Why does my spiral look truncated?
Archimedean and other unbounded spirals keep growing as θ increases. The default 0 to 2π only captures one revolution. For a multi-coil spiral pick 0 to 8π or 0 to 20π from the θ range dropdown — that gives the spiral room to wind several times. The plot auto-scales so the whole curve fits the canvas.
Can I overlay multiple equations?
Yes. Type a second or third equation in the optional input fields. All curves are drawn on the same axes with distinct colors from the active palette. This is ideal for comparing sin(3θ) and cos(3θ), plotting the two halves of a lemniscate, or overlaying a rose inside a cardioid to see how they interact.
What happens if my equation produces negative r?
Negative r is mathematically valid in polar coordinates — it reflects the point through the origin. So r = -1 at θ = 0 is the same as the point r = 1 at θ = π. The plotter handles this correctly, which is why limaçons like r = 1 + 2cos(θ) draw an inner loop where r goes negative.
How can I export the chart?
Three options. Download SVG gives a vector file that stays sharp at any size — perfect for slides, posters, laser cutting and embroidery. Download PNG renders a high-resolution raster up to 1800×1800 pixels, suitable for social media or thumbnails. Copy code puts the raw SVG markup on your clipboard for embedding in a webpage or sending in chat.
Why does the live preview look slightly different from the final result?
The live preview uses 800 samples to stay snappy as you type. The final result uses 600 to 9,000 samples depending on the Resolution dropdown. Both are mathematically equivalent — the higher sample count just produces a smoother stroke, especially on tight curves like dense roses and butterfly spirals.
Is this polar plotter free?
Yes. The Polar Equation Plotter is free, runs entirely in your browser after the form submit, requires no signup and never watermarks the export. Use the charts in homework, papers, slides and commercial projects without restriction.
Reference this content, page, or tool as:
"Polar Equation Plotter" at https://MiniWebtool.com/polar-equation-plotter/ from MiniWebtool, https://MiniWebtool.com/
by miniwebtool team. Updated: 2026-05-21
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