Radius of curvature of r = a(1−cosθ)

Cardioid: r = a(1 − cos θ) = 2a sin²(θ/2). Use the pedal method.

Angle φ: dr/dθ = a sin θ, so

tan φ = r dθ/dr = a(1 − cos θ)/(a sin θ) = tan(θ/2), so φ = θ/2.

Pedal equation: p = r sin φ = r sin(θ/2). From r = 2a sin²(θ/2), sin(θ/2) = √(r/2a). Hence

p = r · √(r/2a) = r^(3/2)/√(2a), that is p² = r³/(2a).

Radius of curvature: ρ = r dr/dp.

dp/dr = (3/2) r^(1/2)/√(2a)

ρ = r · √(2a) / ((3/2) r^(1/2)) = (2/3) √(2a) · √r

Answer: ρ = (2/3)√(2ar) = (4a/3) sin(θ/2).