Beziercurve

Bezier curve, named after the French engineer Pierre Bézier, is a parametric curve widely used in computer graphics and computer-aided design. It is defined by a sequence of control points P0, P1, ..., Pn. The curve is given by B(t) = sum_{i=0}^n B_i^n(t) P_i for t in [0,1], where B_i^n(t) = C(n,i) (1-t)^{n-i} t^i are Bernstein polynomials. The degree n is the number of control points minus one, with quadratic (n=2) and cubic (n=3) forms being the most common.

Evaluation of Bezier curves can be done efficiently with the de Casteljau algorithm, which performs repeated linear interpolations between control points. This method is numerically stable and supports straightforward subdivision of the curve at any parameter value t.

Key properties include B(0) = P0 and B(1) = Pn, so the endpoints lie on the first and last control points. The curve lies within the convex hull of its control points, and the construction is affine-invariant: applying an affine transformation to all control points yields the corresponding transformation of the curve. The tangent directions at the endpoints are given by P1 − P0 and Pn − P_{n−1}, implying that local adjustments near the ends influence the curve's direction there.

Packaging into complex shapes often uses piecewise Bezier segments, typically cubic, with continuity conditions such as C^0, C^1, or higher at joins.

Applications span font outlines, vector graphics, CAD/CAM, and animation. Bezier curves offer intuitive visual editing via control points, but their influence is global to a segment, which can require stitching multiple segments for intricate designs.