Graph and Results
Coordinate Table
| Curve | θ | r | x | y |
|---|
Analysis Summary
No results yet.
Polar Equation Inputs
Formula Used
Polar equations describe distance from the pole at each angle. The calculator samples the equation across the selected interval. Numerical methods estimate area, length, extrema, zeros, and intersections accurately.
How to Use
- Enter one or more expressions for radius.
- Select radians or degrees before setting limits.
- Choose graph styling and coordinate precision.
- Press Plot and Analyse to calculate results.
- Inspect metrics, coordinates, and curve intersections.
- Export the graph, CSV, or printable PDF.
Use smaller steps for smoother curves and improved estimates. Large ranges may require a higher point limit. Keep equations within the supported mathematical functions for reliable results.
Example Data
| Curve | Equation | Suggested Interval | Typical Shape |
|---|---|---|---|
| Circle | r = 3 | 0 to 2π | Circle centred at pole |
| Cardioid | r = 2(1 + cos θ) | 0 to 2π | Heart-shaped loop |
| Rose | r = 3 cos(5θ) | 0 to 2π | Five petals |
| Spiral | r = 0.25 + 0.3θ | 0 to 8π | Expanding spiral |
Frequently Asked Questions
What is a polar equation?
A polar equation gives radius as an angle function. Each angle produces a point from the pole. Connecting sampled points creates the visible polar curve.
Can I enter several equations?
Yes, the calculator compares several enabled equations together. Each curve receives separate styling and analysis. The legend identifies every active curve clearly.
Does it support degrees?
Yes, choose degrees before entering angular limits. Calculations convert degrees internally into radians. Displayed coordinates retain your selected angular unit.
Why are some points missing?
Undefined expressions are skipped instead of breaking calculations. Discontinuities may create visible gaps in curves. Smaller steps can improve nearby curve rendering.
How is enclosed area estimated?
The calculator applies numerical trapezoidal integration to r squared. Results depend on the chosen interval and step. Smaller steps generally improve the estimated area.
How is arc length estimated?
Arc length uses sampled Cartesian distances between neighbouring points. This approach handles many practical polar curves well. Smaller steps usually produce more stable estimates.
What do negative radii mean?
A negative radius plots opposite the current angle direction. This behaviour is valid in polar coordinates. Rose curves often rely on negative radius values.
Can I export coordinates?
Yes, download the visible coordinate data as CSV. The graph can also download as PNG. Use print mode to save results as PDF.
Are symbolic results provided?
The grapher focuses on reliable numerical analysis. It does not promise exact symbolic integration. Displayed results are approximations from sampled equation values.
Accuracy Notice
Numerical results depend on interval size and sampling density. Very small steps increase accuracy but require more processing. Verify important academic or engineering conclusions using trusted methods.