B-Spline Curve with equation

C Graphics » B-Spline Curve with equation

In B-spline curve, Local control is imposed on a curve, means a B-spline curve usually divide into segments and changing control points respective to a particular segment would only change that shape of that region only. In Bezier curve, controls are global, changing a control will lead to change the entire shape of a curve.

Each and every segment uses a unique basis function

S(t) = ∑ni=0   SiNi, p(u)         0<=t<=n-p+2

p are the points that control a segment

S is number of control points

Ni, p(t)  = (u-ki)Ni,p-1(u)/ki+p-1-k (u-ki+p)Ni+1,p-1(u)/ki+p-ki +1

k are the number of knot points

ki where i lies  ( 0<= i <= n+p )

ki = 0,  if i< p

ki = i-p +1 ,  if p<= i<= n

ki = n-p+2,  if i>n

Ni, p(u)     = 1  if  ki <= u <= ki+1

 

Properties of  B-spline curve

1) B-spline curve consists of n+1 control points and p order of the curve.

2) It has local control over curve that controls segments separately.

3) A degree of polynomial depends on the order of the curve which is p-1.

4) B-spline consists of n-p+2 segments.

5) Number of control points can be changed without affecting the degree of a polynomial


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