Polar & Parametric Graph Explorer

Choose a curve type and adjust the parameters to see how the shape changes in real time. Press Trace to watch the curve being drawn point by point. All calculations run in your browser.

Graph

100%

Full curveTraced pathCurrent point

Parameters

Curve Type

a (amplitude)

n (petals)

θ range (×π)

Presets

Curve Info

Equation

r = 4\cos(3\theta)

Properties

Petals3

Max radius4

θ range0 to 2π

Enclosed area25.133

Reference Guide

Polar Coordinates

In polar coordinates, every point is described by a distance $r$ from the origin and an angle $\theta$ from the positive x-axis.

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

When $r$ is negative, the point is plotted in the opposite direction. This is what gives rose curves their petals and limaçons their inner loops.

Polar Curve Gallery

Rose $r = a\cos(n\theta)$ has $n$ petals when $n$ is odd, $2n$ petals when $n$ is even.
Cardioid $r = a(1 + \cos\theta)$ is heart-shaped and passes through the origin.
Limaçon $r = a + b\cos\theta$ has an inner loop when $b > a$ .
Lemniscate $r^2 = a^2\cos(2\theta)$ is a figure-eight.
Spiral $r = b\theta$ winds outward at a constant rate.

Parametric Equations

Parametric equations define x and y each as a function of a third variable $t$ (often representing time).

x = f(t), \quad y = g(t)

As $t$ increases, the point $(x, y)$ traces out a path. The tracing animation shows this process. Parametric form can describe curves that are not functions, like loops and self-intersections.

Area and Arc Length

The area enclosed by a polar curve from $\theta = \alpha$ to $\theta = \beta$ is computed with

A = \frac{1}{2}\int_{\alpha}^{\beta} r^2\, d\theta

The arc length of a parametric curve from $t = a$ to $t = b$ is

L = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2}\, dt

Both are computed numerically using Simpson's rule.