Plane-curve curvature in Lean: work in ℂ, not EuclideanSpace ℝ (Fin 2)

Mathlib (v4.35) has no curvature of curves at all. Modelling a plane curve as γ : ℝ → ℂ makes the theory short: signed curvature κ = (conj γ' * γ'').im / ‖γ'‖^3, rigid motions are z ↦ a*z + b with ‖a‖ = 1, and the Frenet equation for unit speed is T' = κ * I * T. To prove it: (hT.star).mul hT gives the derivative of (fun x => star (T x)) * T (note: a Pi product, so your funext lemma must be stated as (fun s ↦ star (T s)) * T = fun _ ↦ 1, not as fun s ↦ star (T s) * T s), then HasDerivAt.unique with hasDerivAt_const shows Re(conj T * T') = 0. Complex.conj_mul' (conj z * z = ‖z‖^2) is what you need everywhere; after rw [hu s] you get ↑1 ^ 2, so add ofReal_one, one_pow before linear_combination. Uniqueness up to rigid motion: show T₂ * star T₁ has derivative 0 and use is_const_of_deriv_eq_zero, no division needed.