Polar coordinates explained

Beginner tutorial · 10 minute read

Cartesian coordinates ask "how far right and how far up?" Polar coordinates ask a different question: "how far out, and in which direction?" Once that switch clicks, an entire family of beautiful curves becomes natural — flowers, spirals, hearts, lemniscates and limaçons.

The (r, θ) idea

A point in the plane has Cartesian coordinates (x, y): walk x units right, then y units up. The same point also has polar coordinates (r, θ): stand at the origin, rotate your gaze by angle θ from the positive x-axis, then walk forward a distance r.

Conversion is simple:

x = r·cos(θ)
y = r·sin(θ)

A polar equation is a rule that says how far r should be at each angle. The simplest example, r = 1, gives a unit circle: at every angle, you're 1 unit away from the origin.

Your first polar plot

Open the graphing calculator and try:

r = 1 + sin(theta)

You'll see a heart-shaped curve called a cardioid. As θ sweeps from 0 to 2π, the radius rises and falls between 0 and 2, tracing a smooth loop. Try changing the constant to see what happens:

r = 0.5 + sin(theta)
r = 2 + sin(theta)

The rose family

Equations of the form r = a·cos(kθ) or r = a·sin(kθ) create rose curves. The integer k controls the number of petals:

If k is odd, the rose has k petals.
If k is even, the rose has 2k petals.

This is one of those magical asymmetries in mathematics. Try and see:

r = 2*cos(2*theta)
r = 3*sin(5*theta)
r = cos(7*theta)

What if k is a fraction like 5/2? The curve doesn't close after 2π — you have to keep going for several full turns before the pattern repeats. DesmosGraph automatically samples enough θ values to draw these correctly.

Cardioids and limaçons

The general limaçon is r = a + b·cos(θ). The behaviour depends on the ratio of a and b:

a/b > 1: convex limaçon (smooth oval).
a/b = 1: cardioid (heart shape with a cusp).
a/b < 1: limaçon with an inner loop.

Try animating with sliders:

r = a + b*cos(theta)
a = 1
b = 1

Spirals

Two famous spirals:

Archimedes spiral: r = a·θ. Each turn is the same distance further out.
Logarithmic (equiangular) spiral: r = a·e^(b·θ). Each turn is a constant ratio further out — this is the spiral of nautilus shells, galaxies and hurricanes.

r = 0.4*theta
r = exp(0.1*theta)

The lemniscate

The lemniscate of Bernoulli is the figure-eight curve. In polar form: r² = a²·cos(2θ). Because DesmosGraph plots r, we take the positive square root and use abs to handle the negative side:

r = sqrt(abs(cos(2*theta))) * 2

Tips & pitfalls

θ is in radians. If you want degrees, convert: r = cos(theta*pi/180).
Negative radii. When r < 0, the point is reflected through the origin. Many curves use this naturally.
Multiple loops. Polar curves often need θ to range over more than 2π to close. DesmosGraph samples up to 6π by default, which covers the vast majority of common curves.
Discontinuities. If r blows up (e.g. r = 1/cos(θ) at θ = π/2), the plotter automatically breaks the line.

Practice ideas

Plot r = sin(2θ) + cos(3θ) and try changing the integers.
Animate r = a·sin(k·θ) with a slider on k.
Try the butterfly curve: r = exp(sin(θ)) - 2cos(4θ) + sin((2θ-π)/24)^5.

Continue with the implicit equations tutorial for the next step in your visual-math journey.