What is a Bézier curve? How do its defining control points determine its shape…

Concept

A Bézier curve is a parametric polynomial curve defined by a fixed set of control points and Bernstein basis functions. For control points P0, P1, …, Pn, its degree is n and its parameter t runs from 0 to 1.

The Bernstein weights are non-negative and sum to 1, so each point on the curve is a weighted blend of the control points. This gives the curve the convex-hull property and makes each control point exert a smooth “pull” on the path.

Application

1. The curve is B(t) = Σi=0n C(n,i)(1−t)n−itiPi for 0 ≤ t ≤ 1.

2. At t = 0, only the weight of P0 is 1, so the curve starts at P0. At t = 1, only the weight of Pn is 1, so it ends at Pn.

3. The initial tangent is directed along P1 − P0, and the final tangent is directed along Pn − Pn−1. Moving these neighboring control points changes how the curve leaves and approaches its endpoints.

4. Interior control points usually do not lie on the curve. Changing one changes its Bernstein-weighted influence over the whole parameter interval, with the strongest visible pull near the region where its basis function is largest.

5. For the cubic control points (0,0), (1,2), (3,2), and (4,0), the midpoint weights at t = 1/2 are 1/8, 3/8, 3/8, and 1/8. Therefore B(1/2) = (2, 3/2), showing how the two upper interior points lift the path while the endpoints anchor it.

Cross-check

• The Bernstein weights always sum to 1, so B(t) remains inside the convex hull of the control points.

• Reversing the control-point order produces the same geometric path traced in the opposite parameter direction.

• Thus the first and last control points fix the endpoints, adjacent points fix endpoint directions, and all control points collectively determine the curve’s degree, bending, and path.