Polar Graphing Guide — Graph Polar Equations Online

Polar coordinates offer a completely different way to describe position in a plane — using a distance r from the origin and an angle θ (theta) from the positive x-axis. Many beautiful curves that are complicated in Cartesian form become elegantly simple in polar form. Open the graphing calculator to follow this guide.

Understanding Polar Coordinates

In Cartesian coordinates, a point is described by (x, y). In polar coordinates, it's described by (r, θ) where:

• r = the distance from the origin (radius)
• θ = the angle measured counterclockwise from the positive x-axis (in radians)

Conversion between the two systems:

• x = r · cos(θ)
• y = r · sin(θ)
• r = sqrt(x² + y²)

Enabling Polar Mode

In the graphing calculator, open the Settings panel (gear icon in the toolbar) and enable Polar mode. In polar mode, you enter equations in the form r = f(θ) and the calculator plots the curve parametrically as θ sweeps from 0 to 2π.

Basic Polar Curves

Circle

The simplest polar curve is a circle centred at the origin:

r = 3 — a circle with radius 3

Compare with the Cartesian form: x² + y² = 9. The polar form is much simpler!

Rose Curves

Rose curves have petals determined by the coefficient inside the trig function:

• r = cos(2*θ) — 4-petal rose (even n gives 2n petals)
• r = cos(3*θ) — 3-petal rose (odd n gives n petals)
• r = sin(4*θ) — 8-petal rose
• r = cos(5*θ) — 5-petal rose

Cardioid

A cardioid is a heart-shaped curve formed when the radius includes both a constant and trig term of equal magnitude:

• r = 1 + cos(θ) — cardioid pointing right
• r = 1 - sin(θ) — cardioid pointing down
• r = 2 + 2*cos(θ) — larger cardioid

Limaçon

A limaçon is a generalisation of the cardioid where the constant and trig coefficient differ:

• r = 2 + 3*cos(θ) — limaçon with inner loop (if |b| > |a|)
• r = 3 + 2*cos(θ) — convex limaçon (if |a| > |b|)
• r = 2 + 2*cos(θ) — cardioid (when |a| = |b|)

Spirals

Spirals are curves that wind outward from the origin:

• r = θ — Archimedean spiral (evenly spaced coils)
• r = θ / (2*pi) — 1 unit of radius per full rotation
• r = exp(0.2*θ) — logarithmic spiral (found in nature)

Lemniscate

A figure-eight shaped curve:

• r = sqrt(cos(2*θ)) — lemniscate of Bernoulli (only defined where cos(2θ) ≥ 0)
• r^2 = 4*cos(2*θ) — note: r² form; use r = sqrt(...) to approximate

Converting Between Cartesian and Polar

Some Cartesian equations become much simpler in polar form. For example:

• Circle x² + y² = 25 → r = 5
• Line through origin y = x → θ = π/4 (a ray)
• Parabola y = x² → r·sin(θ) = r²·cos²(θ) → complex!

Symmetry in Polar Curves

Polar curves can have beautiful symmetry:

• Symmetric about x-axis: if replacing θ with −θ gives the same equation
• Symmetric about y-axis: if replacing θ with π − θ gives the same equation
• Symmetric about origin: if replacing r with −r gives the same equation

Verify: graph r = cos(θ) — you'll see it's symmetric about the x-axis.