1.5.2—Polar curves

A polar curve is given by r=f(θ). To understand its shape, identify the interval, zeros of r, symmetries and key angles before plotting points.

If f(−θ)=f(θ), the curve is symmetric about the initial line; if f(π−θ)=f(θ), it is symmetric about the line θ=π/2. Negative r plots in the opposite direction, so sign changes are structural.

For r=2cosθ, r is zero at π/2 and negative for π/2<θ<3π/2; the curve is a circle of diameter 2 on the initial-line side after the negative-radius points are interpreted.

Joining a table of positive radii without checking sign and symmetry can produce the wrong curve.