Pedal equation of r² = a²cos2θ

Curve: r² = a² cos 2θ. The pedal equation uses p = r sin φ with tan φ = r dθ/dr.

Take logs: 2 ln r = ln a² + ln cos 2θ. Differentiate with respect to θ:

(2/r)(dr/dθ) = −2 tan 2θ, so r dθ/dr = −cot 2θ = tan(π/2 + 2θ)

Hence φ = π/2 + 2θ, and

p = r sin φ = r cos 2θ.

Since cos 2θ = r²/a²:

p = r³/a²

Pedal equation: p a² = r³.