Graphing Polar Functions

A polar curve is just a point moving around a circle while its distance from the center keeps changing. For each angle theta, you compute r(theta), then mark a point that far from the origin in that direction. Walk theta from 0 to 2pi, plotting as you go, and the points link up into a shape. Where r is large, the curve swings far out; where r shrinks toward zero, the curve pulls in toward the center, sometimes passing right through it.

That shrinking-to-zero behavior is what carves petals and dimples. Take r = 4 sin(2theta): r hits zero whenever 2theta is a multiple of π, meaning theta = 0, π/2, π, 3pi/2. Between each pair of consecutive zeros, r rises from 0 to a max of 4 (or dips to -4) and back to 0, tracing one petal. Four zero-crossings across 0 to 2pi carve four petals. For r = sin(3theta), three petals form the same way, spaced every 2pi/3. A cardioid like r = 2 + 2cos(theta) never hits zero except at theta = π, so instead of petals you get one heart-shaped loop that pinches to a point there.

When r goes negative, the point doesn't vanish, it plots on the opposite ray, at angle theta + π. This flips part of the curve across the origin and can make petals overlap or point in directions you wouldn't expect from the formula alone. This topic is where the algebraic rules for amplitude, period, and symmetry from earlier polar work finally become pictures, and it sets up comparing polar graphs to the rose and circle families you'll classify next.