Bezier spline curves

Linear Bezier spline

Linear Bezier spline is obtained by linear interpolation between two control points P0 , P1
    P(t) = (1-t)P0 + tP1 ,     0 ≤ t ≤ 1
linear interpolation
Interactive Bezier spline   Use finger or mouse to move nearest control point (a small blue square). In the right window you see basis polynomials of the linear Bezier spline. The red line is (1-t) and the green one is t. Images are fitted to your browser window (sorry if your browser doesn't support HTML5).

Quadratic Bezier spline

Quadratic Bezier spline is obtained by deCasteljau algorithm as a linear interpolation between linear interpolation between control points P0 , P1 , P2
    P01 = (1-t)P0 + tP1 ,     P11 = (1-t)P1 + tP2 ,
    P(t) = (1-t)P01 + tP11 = (1-t)[(1-t)P0 + tP1 ] + t[(1-t)P1 + tP2 ] = (1-t)2P0 + 2(1-t)tP1 + t2P2 ,

    P(t) = ∑i=0,2 Bi2(t) Pi ,     Bin(t) = Cni ti(1-t)n-i ,     Cni = n! / i!(n-i)!
where Bin(t) are Bernstein polynomials. To the right you see basis polynomials B02(t) , B12(t) , B22(t) in the red, green, blue, ... order.
deCasteljau algorithm
By construction Bezier spline goes through its terminal control points (P0 , P2 here) and is tangent to the first and last segments of the control polygon. Note that P(t) subdivides the curve in two quadratic splines (see Fig.2). The new curves match the original in position, although they differ in parameterization. Points P0 , P01 , P(t) and P(t) , P11 , P1 are control points of the new small splines .

Cubic Bezier spline

In a similar way one can use deCasteljau algorithm to interpolate a curve between (n+1) control points P0 , P1 ... Pn . E.g. for n = 3 we get cubic Bezier spline
deCasteljau algorithm
Points P0 , P01 , P02 , P(t) and P(t) , P12 , P21 , P3 are control points of new small splines again. The control points of the two new curves appear along the sides of the systolic array (see Fig.4 below).

How to plot Bezier spline

DeCasteljau iterations
    Pij = (1-t)Pij-1 + tPi+1j-1,     j = 1, n     i = 0, n-j
for n = 3 are shown on the scheme in Fig.4
Fig.4   Fig.5     Fig.5a
This algorithm is programmed as (see bezier.js too)
   for (j = N-1; j > 0; j--)
    for (i = 0; i < j; i++){
      Px[i] = (1-t)*Px[i] + t*Px[i+1];
      Py[i] = (1-t)*Py[i] + t*Py[i+1];
where N = n+1 is number of control points.
To obtain basis polynomials recurrence relations can be used
    B00(t) = 1,
    Bin(t) = (1-t)Bin-1(t) + tBi-1n-1(t),

where Brs = 0 when r < 0 or r > s.
This algorithm is shown on Fig.5a . It is programmed as
   B[1] = 1;
   for (j = 1; j < N; j++)
    for (i = j+1; i > 0; i--)
      B[i] = (1-t)*B[i] + t*B[i-1];
here B[0] = 0 is added for convenience, therefore i -> i+1. Basis functions can be obtained too without "zeros" (see Fig.5)
   B[0] = 1;
   for (j = 0; j < n; j++)
    for (i = j; i => 0; i--){
      B[i+1] += t*B[i];
      B[i] += (1-t)*B[i];
Bezier spline of degree n = 7

The general expression for the Bezier spline of degree n (order n+1) is
    P(t) = ∑i=0,n Bin(t) Pi .
Further you can read More Bezier splines Math or go directly to Interpolating Lagrange curve.
Contents   Next: More Bezier splines Math
updated 7 August 2001