purpose

Polar to Cartesian Converter & Grapher

Convert polar equations to Cartesian coordinates and visualize both representations. See r = f(\u03b8) plotted alongside x, y coordinates.

Back to Polar Coordinate Plotter | Interactive Polar Grapher

Curves (1/6)

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Display

Cartesian Grid

Axes

Polar Grid

Presets

AllRosesCardioids & LimaçonsSpiralsLemniscatesConicsCirclesSpecial

Documentation Contents

How to Use This Plotter

Enter polar equations of the form r = f(theta) and see them plotted instantly on an interactive canvas.

Enter an expression: Type your polar equation using theta as the angle variable. For example, 1 + cos(theta) produces a cardioid. Standard math functions (sin, cos, sqrt, abs, exp, log) and constants (pi, e) are supported.
Adjust the theta range: The default range is $[0, 2\pi]$ . Roses with odd $n$ complete in $[0, \pi]$ , while spirals need $[0, 4\pi]$ or more to show multiple turns. Use the quick-range buttons or type custom values.
Add more curves: Click the Add button to plot up to 6 curves simultaneously, each with its own color, expression, and theta range.
Explore presets: The preset gallery contains 20+ famous polar curves organized by family (roses, cardioids, spirals, lemniscates, conics, circles, and special curves). Click any preset to load it instantly.
Animate the trace: Use the play button to watch how the curve is traced as theta sweeps through its range. Pause at any point to study the curve at a specific angle.

Tip

Negative $r$ values are handled automatically using the standard polar convention: the point $(r, \theta)$ with $r < 0$ is plotted at $(|r|, \theta + \pi)$ , reflecting it through the origin.

The Polar Coordinate System

In polar coordinates, every point in the plane is described by two values:

$r$ (radius): the distance from the origin (called the pole)
$\theta$ (theta): the angle measured counterclockwise from the positive x-axis (called the polar axis)

Conversion Formulas

To convert between polar and Cartesian coordinates:

x = r\cos\theta, \quad y = r\sin\theta

r = \sqrt{x^2 + y^2}, \quad \theta = \arctan\!\left(\frac{y}{x}\right)

Unlike Cartesian coordinates, polar representations are not unique. The same point can be written as $(r, \theta)$ , $(r, \theta + 2\pi)$ , or $(-r, \theta + \pi)$ .

Polar Curve Families

Roses (Rhodonea)

r = a\cos(n\theta) \quad \text{or} \quad r = a\sin(n\theta)

Produces petal-like patterns. If $n$ is odd, the curve has $n$ petals over $[0, \pi]$ . If $n$ is even, it has $2n$ petals over $[0, 2\pi]$ . The maximum radius is $|a|$ .

Cardioids & Limaçons

r = a + b\cos\theta \quad \text{or} \quad r = a + b\sin\theta

When $a = b$ , the curve is a cardioid (heart shape). When $a < b$ , a limaçon with inner loop forms. When $a > b$ , the limaçon is dimpled or convex.

Spirals

r = a + b\theta \quad (\text{Archimedean})

r = ae^{b\theta} \quad (\text{Logarithmic})

Archimedean spirals expand at a constant rate. Logarithmic spirals grow exponentially and appear in nautilus shells, galaxy arms, and hurricane patterns.

Lemniscates

r^2 = a^2\cos(2\theta)

Figure-eight curves discovered by Jacob Bernoulli in 1694. The lemniscate of Bernoulli is the inspiration for the infinity symbol ( $\infty$ ).

Conic Sections

r = \frac{ed}{1 + e\cos\theta}

When $e < 1$ , the conic is an ellipse. When $e = 1$ , a parabola. When $e > 1$ , a hyperbola. This is the form used inKepler's laws for planetary orbits.

Circles

r = a \quad \text{or} \quad r = 2a\cos\theta

$r = a$ is a circle of radius $a$ centered at the origin. $r = 2a\cos\theta$ is a circle of radius $a$ tangent to the origin at the right.

Key Formulas

Polar Area

The area enclosed by a polar curve $r = f(\theta)$ from $\alpha$ to $\beta$ is:

A = \frac{1}{2}\int_{\alpha}^{\beta} r^2 \, d\theta = \frac{1}{2}\int_{\alpha}^{\beta} [f(\theta)]^2 \, d\theta

This formula works because each infinitesimal sector has area $\tfrac{1}{2}r^2 \, d\theta$ , which is the polar analogue of the rectangular area element.

Arc Length

The arc length of a polar curve from $\alpha$ to $\beta$ is:

L = \int_{\alpha}^{\beta} \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2} \, d\theta

The derivative $dr/d\theta$ is computed numerically using a central difference approximation.

Slope of a Polar Curve

To find the slope $dy/dx$ at a point on a polar curve:

\frac{dy}{dx} = \frac{\frac{dr}{d\theta}\sin\theta + r\cos\theta}{\frac{dr}{d\theta}\cos\theta - r\sin\theta}

Symmetry Tests

Polar axis (x-axis): $f(-\theta) = f(\theta)$ -- replacing $\theta$ with $-\theta$ gives the same equation.
$\pi/2$ axis (y-axis): $f(\pi - \theta) = f(\theta)$
Origin: $f(\theta + \pi) = -f(\theta)$ -- equivalent to $f(-\theta) = -f(\theta)$

Real-World Examples

Example 1: Microphone Polar Patterns

Audio engineers use polar plots to visualize how a microphone picks up sound from different directions. The three most common patterns are:

Omnidirectional: $r = 1$ -- picks up sound equally from all directions.
Cardioid: $r = 0.5(1 + \cos\theta)$ -- rejects sound from behind ( $\theta = \pi$ ). This is the standard vocal microphone pattern.
Figure-8 (bidirectional): $r = |\cos\theta|$ -- picks up from front and back but rejects from the sides.

Concrete numbers: At $\theta = 90°$ (directly to the side), the cardioid sensitivity is $r = 0.5(1 + \cos 90°) = 0.5$ , which is 6 dB quieter than the front ( $r = 1.0$ ). At $\theta = 180°$ (directly behind), the sensitivity drops to $r = 0.5(1 + \cos 180°) = 0$ , providing complete rejection.

Takeaway: Plot both patterns simultaneously to compare their directional response visually.

Example 2: Antenna Radiation Patterns

Radio engineers plot antenna gain in polar coordinates to visualize the radiation pattern. A simple dipole antenna has a figure-eight pattern, while more complex arrays produce rose-like patterns with controlled beamwidth.

Equation: A 3-element array pattern can be approximated as $r = |\cos(3\theta)|$ .

Concrete numbers: The main lobe of $r = |\cos(3\theta)|$ has its peak at $\theta = 0$ with $r = 1$ (0 dBi gain). The first null occurs at $\theta = \pi/6 = 30°$ where $r = |\cos(90°)| = 0$ . The 3 dB beamwidth (where $r = 1/\sqrt{2} \approx 0.707$ ) is approximately $2 \times 15° = 30°$ .

Takeaway: The number of lobes in the radiation pattern corresponds to the frequency parameter in the polar equation.

Example 3: Kepler's Planetary Orbits

Kepler's first law states that planetary orbits are ellipses with the Sun at one focus. In polar coordinates with the Sun at the origin:

r = \frac{a(1 - e^2)}{1 + e\cos\theta}

Concrete numbers for Earth: Semi-major axis $a = 1.000\,\text{AU}$ , eccentricity $e = 0.0167$ .

Perihelion ( $\theta = 0$ ): $r = \frac{1.000(1 - 0.0167^2)}{1 + 0.0167} \approx 0.983\,\text{AU}$ = 147.1 million km
Aphelion ( $\theta = \pi$ ): $r = \frac{1.000(1 - 0.0167^2)}{1 - 0.0167} \approx 1.017\,\text{AU}$ = 152.1 million km

For Mars ( $e = 0.093$ , $a = 1.524\,\text{AU}$ ): perihelion = $1.381\,\text{AU}$ , aphelion = $1.666\,\text{AU}$ . The higher eccentricity makes the ellipse visibly more elongated than Earth's.

Takeaway: Plot both orbits on the same graph using multi-curve mode to compare the eccentricity difference visually.

Example 4: Logarithmic Spirals in Nature

The nautilus shell, hurricane arms, and spiral galaxies all follow a logarithmic spiral pattern where the growth rate is constant:

r = ae^{b\theta}

Concrete numbers for a nautilus shell: With $a = 1$ and $b = 0.175$ (corresponding to a growth factor of about 3 per full revolution):

At $\theta = 0$ : $r = e^0 = 1.0$ (1 mm reference)
At $\theta = 2\pi$ : $r = e^{0.175 \times 6.28} \approx 3.0$ (3 mm)
At $\theta = 4\pi$ : $r = e^{0.175 \times 12.57} \approx 9.0$ (9 mm)
At $\theta = 6\pi$ : $r = e^{0.175 \times 18.85} \approx 27.1$ (27 mm)

Each revolution triples the radius -- a hallmark of self-similar growth. Use the plotter with e^(0.175 * theta) and a theta range of $[0, 6\pi]$ to see the spiral.

Takeaway: The logarithmic spiral maintains a constant angle between the curve and radial lines -- this property is called equiangularity.

Common Mistakes

Wrong theta range for roses: A rose curve $r = a\cos(n\theta)$ with odd $n$ completes in $[0, \pi]$ , not $[0, 2\pi]$ . Using $[0, 2\pi]$ traces every petal twice, which doubles the computed area. For even $n$ , the range $[0, 2\pi]$ is correct.
Confusing degrees and radians: The plotter works in radians. If you enter cos(45) expecting $\cos 45°$ , you will get $\cos(45\,\text{rad})$ instead. Convert degrees to radians: $45° = \pi/4 \approx 0.785\,\text{rad}$ .
Forgetting that negative r reflects through the origin: When $r < 0$ , the point is plotted at $(|r|, \theta + \pi)$ . This is how rose curves create petals on opposite sides of the origin.
Insufficient theta range for spirals: An Archimedean spiral $r = \theta/(2\pi)$ needs $[0, 4\pi]$ for 2 turns or $[0, 6\pi]$ for 3 turns. The default $[0, 2\pi]$ shows only one revolution.
Double-counting area for curves with negative r: The area formula $\tfrac{1}{2}\!\int r^2\,d\theta$ uses $r^2$ , which is always positive. For curves like $r = \cos(2\theta)$ that go negative, the formula counts the area of the reflected petals as well.

Frequently Asked Questions

What variable name should I use for the angle?

Use theta for the angle variable and pi for the constant $\pi$ . For example, 1 + cos(theta) for a cardioid, cos(2 * theta) for a 4-petal rose, or theta / (2 * pi) for an Archimedean spiral.

How accurate are the area and arc length calculations?

The area and arc length are computed using numerical integration with 2000 sample points (trapezoidal rule). For smooth, well-behaved curves, this gives accuracy to about 3-4 decimal places. For curves with sharp features or rapid oscillations, the accuracy may be lower.

Can I plot curves with negative r values?

Yes. Negative $r$ values are handled using the standard polar convention: the point $(r, \theta)$ with $r < 0$ is plotted at $(|r|, \theta + \pi)$ . This is how rose curves like $r = \cos(2\theta)$ create petals that extend in all directions.

Why does my curve look different from my textbook's graph?

Common causes: (1) Different theta range -- check whether your textbook uses $[0, \pi]$ or $[0, 2\pi]$ . (2) Sine vs cosine -- $r = \cos\theta$ and $r = \sin\theta$ produce the same shape but rotated by $90°$ . (3) The plotter uses radians, so ensure your inputs are in radians.

Can I save or share my plots?

The current configuration is encoded in the URL. Copy the URL to share your exact setup, including all curves, theta ranges, and display settings. You can also use the screenshot button in the graph toolbar to save an image.

What math functions are supported?

The expression parser supports all standard functions: sin, cos, tan, sec, csc, cot, asin, acos, atan, exp, log (natural), log10, sqrt, cbrt, abs, floor, ceil, round, sign, and the ^ operator for exponents. Constants: pi and e.

Is this plotter free to use?

Yes, completely free with no paywalls, sign-ups, or ads. All features including animated tracing, multi-curve plotting, and curve analysis are available at no cost.

References

Stewart, J. Calculus: Early Transcendentals, 9th ed., Cengage, 2020. Sections 10.3–10.6.
Wikipedia. "Polar coordinate system." https://en.wikipedia.org/wiki/Polar_coordinate_system
Wolfram MathWorld. "Rose Curve." https://mathworld.wolfram.com/RoseCurve.html
MacTutor History of Mathematics. "Coolidge: Origin of Polar Coordinates." University of St Andrews. https://mathshistory.st-andrews.ac.uk/Extras/Coolidge_Polars/
Mathematics LibreTexts. "Polar Coordinates: Graphs." Trigonometry (OpenStax). https://math.libretexts.org/Courses/Las_Positas_College/Math_39:_Trigonometry/04:_Further_Applications_of_Trigonometry/4.03:_Polar_Coordinates_-_Graphs

Disclaimer

This polar coordinate plotter is provided for educational and informational purposes only. While it uses mathematically rigorous numerical methods for area and arc length computation, floating-point arithmetic can introduce small rounding errors. The area and arc length values are approximations computed via numerical integration with a fixed resolution, not exact symbolic solutions. Curve family detection and symmetry analysis use heuristic methods that may occasionally misclassify exotic or degenerate curves. Always verify critical results independently. Do not rely on this tool for safety-critical engineering calculations without independent verification by a qualified professional.

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