B-spline basis functions

这篇可能帮我们更好的理解B-spline。

The equation for k-order B-spline with n+1 control points (P₀ , P₁ , ... , P_n ) is
    P(t) = ∑_i=0,n N_i,k(t) P_i ,     t_k-1 <= t <= t_n+1 .
In a B-spline each control point is associated with a basis function N_i,k which is given by the recurrence relations (see Bspline.java)
    N_i,k(t) = N_i,k-1(t) (t - t_i)/(t_i+k-1 - t_i) + N_i+1,k-1(t) (t_i+k - t)/(t_i+k - t_i+1) ,
    N_i,1 = {1   if   t_i <= t <= t_i+1 ,    0   otherwise }
N_i,k is a polynomial of order k (degree k-1) on each interval t_i < t < t_i+1. k must be at least 2 (linear) and can be not more, than n+1 (the number of control points). A knot vector (t₀ , t₁ , ... , t_n+k) must be specified. Across the knots basis functions are C ^k-2 continuous.
   
Corresponding iterations scheme for cubic ( k = 4) basis functions is shown in Fig.1 . You see, that for a given t value only k basis functions are non zero, therefore B-spline depends on k nearest control points at any point t.

B-spline basis functions as like as Bezier ones are nonnegative N_i,k >= 0 and have "partition of unity" property
    ∑_i=0,n N_i,k(t) = 1,     t_k-1 < t < t_n+1
therefore
    0 <= N_i,k <= 1.
As since N_i,k = 0 for t <= t_i or t >= t_i+k therefore a control point P_i influences the curve only for t_i < t < t_i+k.