| θ (deg) | r | x | y |
|---|---|---|---|
| 0 | 0.000 | 0.000 | 0.000 |
| 10 | 1.000 | 0.985 | 0.174 |
| 20 | 1.732 | 1.628 | 0.592 |
| 30 | 2.000 | 1.732 | 1.000 |
| 40 | 1.732 | 1.327 | 1.114 |
| 50 | 1.000 | 0.643 | 0.766 |
- Polar equation: you define r = f(θ).
- Cartesian conversion: x = r·cos(θ), y = r·sin(θ).
- Area (approx): A = ½ ∫ r(θ)² dθ, estimated by the trapezoidal rule.
- Curve length (approx): computed from successive points: Σ √((Δx)²+(Δy)²).
- Negative r: if reflected, use r = |r| and θ = θ + π.
- Enter up to three equations, then enable the curves.
- Pick degrees or radians, then set the θ range.
- Choose a step size and adjust the point limit.
- Set parameters a–d if your equations use them.
- Fine-tune plot controls like direction and zero location.
- Click Plot, then export CSV or PDF.
1) What a polar plot represents
A polar curve assigns a radius r to each angle θ. Each sampled pair (r, θ) becomes a point after conversion: x = r·cos(θ) and y = r·sin(θ). Common examples include roses r = 2·sin(5θ), cardioids r = 2(1 − cos θ), and Archimedean spirals r = kθ.
2) Equation entry rules
Write r(θ) using the variable theta. You can use functions such as sin, cos, tan, sqrt, abs, and powers. Constants pi and e are available, plus parameters a, b, c, and d for quick “what-if” changes without rewriting the formula.
3) Range, step, and point limits
The calculator samples θ from start to end using a fixed step, creating about |end − start|/step + 1 rows per curve. Smaller steps usually look smoother and improve numeric estimates, but they increase runtime and export size. A maximum point limit (default 50,000) prevents accidental heavy plots.
4) Negative radius options
Some equations produce negative r. “Reflect” applies the standard polar convention by flipping r to |r| and shifting θ by π, placing the point on the opposite ray. “Keep” preserves negative r values for debugging. “Clip” forces r to zero to restrict the graph to nonnegative radii.
5) Overlay up to three curves
You can display three equations on one plot to compare shapes or parameters. Each curve has a label and line settings (dash, width, opacity) plus marker mode. For example, compare r = a(1 − cos θ) for multiple a values, or overlay a rose curve with a limaçon.
6) Metrics computed from samples
For each curve, area is estimated by A ≈ ½∫r(θ)² dθ using the trapezoidal rule over the sampled angles (in radians). Curve length is estimated by summing straight-line distances between successive points: Σ√((Δx)²+(Δy)²). Reduce step size if the curve has tight loops.
7) Export and reporting workflow
The CSV export includes θ in your chosen unit, θ in radians, and r/x/y columns for each enabled curve on the same θ grid. The PDF export embeds the plot image, summaries, and a preview table, so your settings and results can be shared or reproduced later. Use manual radial limits when you want consistent scaling across multiple comparisons runs.
1) What variable name should I use for the angle?
Use theta inside your equation. The calculator evaluates r from that variable and then converts to x and y for plotting and exports.
2) Why does a degree input still use radians in the equation?
Trig functions are evaluated in radians. When you choose degrees, θ is converted to radians before the expression is computed, so formulas behave consistently.
3) My curve looks jagged. How do I smooth it?
Reduce the step size to sample more points. If you exceed the max-points limit, widen the step slightly or shorten the θ range until the plot is stable.
4) What does “Reflect” do for negative r?
It replaces r with |r| and adds π to θ, which moves points to the opposite direction. This matches the usual polar interpretation of a negative radius.
5) Are area and curve length exact?
No. They are numerical estimates based on sampling. Smaller step sizes typically improve accuracy, but very sharp features may still need additional refinement or analytic methods.
6) Can I export all curves at once?
Yes. The CSV export writes columns for every enabled curve on the same θ grid. The PDF summarizes enabled curves and embeds the plotted image.