Bezeir blending functions.jpg

Hi,

I know that By Blending function we mean function which produces curve which can fit between two control points. The figure shows me Bezeir curve consisting of 4 control Points P0, P1, P2, and P3 and 4 blending functions. Two blending functions are labeled as P0 and P3 which are interpolation control points where as two points are labeled as P1 and P2 which are approximating control points. I don’t know how we can have one to one correspondence between portions of Bezeir curves and Blending functions.

I got following explanation from a web site:

I can understand that at u=0 and at u=1 we have blending functions having value 1, but why they are labeled as P0 and P3 which are interpolating control points. The shape of Blending function labeled P3 is same as the portion of Bezeir curve but the shape of P0 is not same as the portion of Bezeir curve. Same is true for blending curves P1 and P2. The shape of curve P2 is different?At u=0, the only nonzero blending function is p0 which has the value 1. At u=1, the only non-zero function is p3 with a value of 1 at that point. Thus the cubic Bezeir curve will always pass through the control points p0 and p3 . The other functions , p1 and p2 influence the shape of curve at intermediate values of parameter u, so that the resulting curve tends toward points p1 and p2. Blending function p1 is maximum at u=1/3 and p2 is maximum at u= 2/3.

Plz guide about how to choose blending functions to fit between control points?