Defining Data

A degree-n Bezier curve is defined by n+1 control points P0, P1, ..., Pn: P(u) = sum over i=0..n of B_(i,n)(u) P_i, u in [0, 1], where B_(i,n)(u) = C(n, i) u^i * (1-u)^(n-i) is the Bernstein polynomial.

Cubic Bezier (n=3)

P(u) = (1-u)^3 P0 + 3(1-u)^2 u P1 + 3(1-u) u^2 P2 + u^3 P3.

Properties

1. Endpoint interpolation: P(0) = P0, P(1) = P3.
2. Tangent at endpoints: P'(0) = 3(P1 - P0), P'(1) = 3(P3 - P2). The curve leaves P0 toward P1 and arrives at P3 from P2.
3. Convex hull: the entire curve lies inside the convex hull of its control points.
4. Affine invariance: applying an affine transform to the control points is the same as applying it to the curve.
5. Variation-diminishing: a straight line crosses the curve no more times than it crosses the control polygon.
6. Partition of unity: sum of B_(i,n)(u) = 1 for all u, so any control-point combination is an affine combination.

de Casteljau Algorithm

A numerically stable, geometrically intuitive way to evaluate P(u) for a given u:

1. P_i^0 = P_i for i=0..n.
2. For r=1..n: P_i^r(u) = (1-u) P_i^(r-1)(u) + u P_(i+1)^(r-1)(u), for i=0..n-r.
3. P_0^n(u) = P(u).

It also splits the curve at u into two Bezier sub-curves whose control points are by-products of the algorithm.

Worked Example

Cubic with P0 = (0,0), P1 = (1,2), P2 = (3,2), P3 = (4,0). Evaluate at u=0.5:

• Layer 1: Q0 = 0.5(0,0)+0.5(1,2) = (0.5,1); Q1 = 0.5(1,2)+0.5(3,2) = (2,2); Q2 = 0.5(3,2)+0.5(4,0)=(3.5,1).
• Layer 2: R0 = 0.5Q0+0.5Q1 = (1.25, 1.5); R1 = 0.5Q1+0.5Q2 = (2.75, 1.5).
• Layer 3: P(0.5) = 0.5R0 + 0.5R1 = (2, 1.5).

Why Bezier Wins in Design

• Drag a control point - curve responds locally to leg P_(i-1)-P_i-P_(i+1) and globally is bounded by the hull, so artists predict shape change well.
• Used in fonts (TrueType uses quadratic Bezier; PostScript and CFF use cubic), Illustrator paths, Inkscape, SVG.

Joining Bezier Segments

For C1 across two cubic Bezier segments meeting at a shared endpoint S, the leg lengths and direction must match: the previous segment's last leg and the next segment's first leg are colinear and equal-length. Many illustration tools call this a smooth handle.