Pedal equation of r^m = a^m cos mθ

The pedal equation is a relation between r and p, where p is the perpendicular from the pole to the tangent. Use p = r sin φ, where tan φ = r dθ/dr.

Take logs of r^m = a^m cos mθ:

m ln r = m ln a + ln cos mθ

Differentiate with respect to θ:

(m/r)(dr/dθ) = −m tan mθ, so (1/r)(dr/dθ) = −tan mθ

tan φ = r dθ/dr = −cot mθ = tan(π/2 + mθ)
so   φ = π/2 + mθ

Then

p = r sin φ = r sin(π/2 + mθ) = r cos mθ

From the curve, cos mθ = r^m / a^m. Hence

p = r^(m+1) / a^m

Pedal equation: a^m p = r^(m+1).