Geometric characteristics of conics in Bézier form

In the Bézier formalism, an arc of a conic is a rational curve of degree 2 with control polygon {P, Q, R} for which the weights can be normalized to {1, w, 1}. The parametrization of the conic arc is C(t) = (1 − t) 2 P + 2wt(1 − t)Q + t 2 R (1 − t) 2 + 2wt(1 − t) + t 2 , t ∈ [0, 1]. Abstract Synthetic derivation of closed for-mulae of the geometric characteristic of a conic given in Bézier form in terms of its control polygon, Notation Consider the frame, {P, Q, R}, defined by the points of the control polygon of the conic. Let (α, β, γ) ∈ P 2 be the coordinates of a point in this frame. The conic lies on the affine plane defined by points whose coordinates satisfy α + β + γ = 1. Points on the line at infinity, z, satisfy α + β + γ = 0. A point in z is a direction on the affine plane. The dual frame, {p, q, r}, of linear forms is associated to the polar lines of {P, Q, R}, normalized so that 1 = p(R) = q(Q) = r(P). In this frame, lines on the affine plane have coordinates (π, ρ, σ).