Interpolating cubic B-spline

Bezier control points

B-spline does not interpolate its deBoor control points. As since Bezier curve goes through its terminal points therefore we will use Bezier control points for cubic uniform B-spline (really we use here only cubic Bezier splines joined C² smoothly).

To determine the first cubic Bezier segment we could set its two terminal control points and tangents. Then the second segment is determined by the first point, two derivatives (continuous at the point) and the second control point. All the rest segments are determined in a similar way.

It is more convinient to set terminal tangents of the whole spline curve. Fig.1' shows, that all Bezier control points of the curve are evaluated from two vectors (P₀, P₁, ... , P_n) and (d₀, d₁, ... , d_n). Points (P₀, P₁, ... , P_n), d₀, d_n are set by user and d₁, d₂, ... , d_n-1 are calculated.

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Interactive interpolating cubic B-spline Drag/Add/Delete control points by finger or mouse.

Solving banded equations

It is evident that by definition (see Fig.1) the curve is C¹ continuous. From C² continuity it follows e.g.
P₁ - 2(P₁ - d₁) + (P₀ + d₀) = (P₂ - d₂) - 2(P₁ + d₁) + P₁ ,
d₀ + 4d₁ + d₂ = P₂ - P₀ .
In general case we get banded 3-diagonals linear equations
4d₁ + d₂ = P₂ - P₀ - d₀
d₁ + 4d₂ + d₃ = P₃ - P₁
... ...
d_i + 4d_i+1 + d_i+2 = P_i+2 - P_i
... ...
d_n-2 + 4d_n-1 = P_n - P_n-2 - d_n
We can successively express
d_i = A_i + B_i d_i+1 , i = 1 , 2 , ... , n-1
where
A₁ = (P₂ - P₀ - d₀)/4 , B₂ = -1/4,
A_i = (P_i+1 - P_i-1 - A_i-1)/(4 + B_i-1), B_i = -1/(4 + B_i-1).
Then we will find d_n-1 and d_i successively by iterations in reverse order
d_n-1 = A_n-1 + B_n-1 d_n, d_i = ... .

It is programmed as (b-interpolate.js)

function findCPoints(){
  Bi[1] = -.25;
  Ax[1] = (Px[2] - Px[0] - dx[0])/4;   Ay[1] = (Py[2] - Py[0] - dy[0])/4;
  for (var i = 2; i < N-1; i++){
   Bi[i] = -1/(4 + Bi[i-1]);
   Ax[i] = -(Px[i+1] - Px[i-1] - Ax[i-1])*Bi[i];
   Ay[i] = -(Py[i+1] - Py[i-1] - Ay[i-1])*Bi[i]; }
  for (var i = N-2; i > 0; i--){
   dx[i] = Ax[i] + dx[i+1]*Bi[i];  dy[i] = Ay[i] + dy[i+1]*Bi[i]; }
}

Handling the terminal tangents

If you do not like to worry about the terminal tangents of interpolating curve, you can direct them to the nearest point as for Cardinal curve, i.e. ("1/3 rule" is used)
d₀ = (P₁ - P₀)/3, d_n = (P_n - P_n-1)/3.

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Contents Previous: Nonuniform B-splines Next: NURBS
updated 21 August 2001