cubicbezier

Cubic Bézier curves are parametric curves defined by four control points P0, P1, P2, and P3 in the plane or in space. The standard parametric form is B(t) = (1−t)^3 P0 + 3(1−t)^2 t P1 + 3(1−t) t^2 P2 + t^3 P3 for t in the interval [0, 1]. This ensures that B(0) = P0 and B(1) = P3, with the initial tangent directed toward P1 from P0 and the final tangent directed toward P3 from P2.

The curve can also be expressed via Bernstein polynomials: B(t) = ∑_{i=0}^3 B_i^3(t) P_i, where B_i^3(t) = C(3,i)(1−t)^{3−i} t^i. This form highlights the influence of each control point on the shape of the curve. The cubic Bézier curve is contained within the convex hull of its four control points and is affine-invariant, meaning it behaves predictably under linear transformations.

Practical computation often uses the De Casteljau algorithm, a stable, recursive method to evaluate B(t) and to subdivide the curve into shorter cubic segments. Cubic Béziers are widely used in vector graphics, font design, and computer-aided design, and they underpin many animation easing functions, such as those defined by CSS cubic-bezier specifications.

Compared with quadratic Béziers, cubic Béziers offer greater flexibility due to the additional control point. However, the parameter t does not correspond to arc length, so motion along the curve may be nonuniform unless remapped.