Results
Ready for inputInteractive polar graph
Curve analysis
Coordinate table
| # | Equation | θ | r | x | y |
|---|---|---|---|---|---|
| No coordinate data yet. | |||||
Merged calculation report
No report generated yet.
Formula used
Polar points use x = r cos(θ) and y = r sin(θ). Cartesian points use r = √(x² + y²) and θ = atan2(y, x). These formulas preserve quadrant information.
Polar area uses A = ½∫r²dθ across the selected interval. Polar arc length uses L = ∫√(r² + (dr/dθ)²)dθ. Numerical integration estimates both values.
The tangent slope is dy/dx = (r′sinθ + rcosθ) ÷ (r′cosθ − rsinθ). The calculator estimates r′ with a centred difference. Undefined vertical tangents are reported clearly.
How to use
Select a preset or type a valid equation. Choose the angle mode and interval. Add more equations when comparison is required.
Adjust graph, precision, and integration controls. Press Calculate and graph to create results. Drag the graph or use zoom controls.
Review analysis and coordinate tables. Export CSV, PDF, PNG, or copied text. Use point evaluation for tangent details.
Example data
| Curve | Equation | Suggested interval | Expected shape |
|---|---|---|---|
| Circle | r = 4cos(θ) | 0° to 360° | Circle through the pole |
| Cardioid | r = 3(1 + cos(θ)) | 0° to 360° | Heart-shaped loop |
| Rose | r = 4cos(5θ) | 0° to 360° | Five petals |
| Spiral | r = 0.35θ | 0° to 1080° | Expanding spiral |
Practical applications
Polar graphs describe rotating systems and directional measurements. Engineers use them for antennas and radiation patterns. Physicists use them for orbital and wave models.
Navigation applications represent bearings and radial distances. Robotics uses polar data from range sensors. Computer graphics uses polar transformations for procedural shapes.
Students compare algebraic equations with visual behaviour. Numerical tools reveal areas and curve lengths. Exported coordinates support further spreadsheet analysis.
Frequently asked questions
What is a polar coordinate?
A polar coordinate describes a point using radius and angle. Radius measures distance from the pole. Angle measures direction from the polar axis.
Can the calculator graph negative radius values?
Yes. A negative radius places the point in the opposite direction. This behaviour is preserved during plotting.
Can I enter more than one equation?
Yes. Add equation rows for comparisons. Each curve has independent colour, width, style, and visibility controls.
Why does a graph contain gaps?
Gaps usually indicate undefined or extremely large values. The graph separates discontinuities to avoid false connecting lines.
How is enclosed area calculated?
The calculator numerically integrates one-half of radius squared. Results depend on interval choice and sample precision.
How is polar arc length calculated?
Arc length combines radius and its angular derivative. The calculator estimates the derivative and numerically integrates the result.
Which angle modes are supported?
Degrees, radians, and gradians are supported. Internally, calculations are converted into radians for consistent evaluation.
Can I export the plotted coordinates?
Yes. Download the coordinate table as CSV. You can also export PDF, PNG, or copied text results.
Are area and period results exact?
They are numerical estimates. Increasing integration intervals usually improves accuracy. Symbolic simplification is not performed.