Radius of curvature of the astroid

Astroid: x^(2/3) + y^(2/3) = a^(2/3), with x = a cos³t, y = a sin³t.

Method 1 (parametric):

x′ = −3a cos²t sin t,   y′ = 3a sin²t cos t
x″ = 3a cos t (3 sin²t − 1),   y″ = 3a sin t (3 cos²t − 1)
x′² + y′² = 9a² sin²t cos²t
x′y″ − y′x″ = −9a² sin²t cos²t

|ρ| = (x′² + y′²)^(3/2) / |x′y″ − y′x″| = 27a³ sin³t cos³t / (9a² sin²t cos²t) = 3a sin t cos t.

Method 2 (pedal): from the pedal equation r² = a² − 3p², differentiate: 2r dr = −6p dp, so r dr/dp = −3p, hence ρ = r dr/dp = −3p, and |ρ| = 3p = 3a sin t cos t.

In terms of x, y: xy = a² sin³t cos³t, so a sin t cos t = (a x y)^(1/3).

Answer: ρ = 3 (a x y)^(1/3) in magnitude (= 3a sin t cos t).