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If the number of control points is n+1, and the degree of the basis function is p

If n = p, B-spline is as same as Bézier curve.

Suppose I have a chance to increase the number of control points say to be n+2; What advantage I can get by doing so compared to Bézier.

Thank you very much

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With degree $p$, a bezier can only have $p+1$ control-points. For a composite bezier curve the number of control points must be a multiple $m$ of $p+1$.
With a B-spline you can increase the number of parameters by $1$.

When using $d$-dimensional composite bezier curves, the control points are usually constrained to obtain $G^1$-continuity. This reduces the dimension of the parameter space from $dm(p+1)$ to $dm(p-1)+m$.

Replacing such a bezier by a B-spline is the same as eliminating some of the parameters by further constraining the derivatives of approximation to be continuous. This reduction of free parameters makes the optimization less computationally heavy.

If the eliminated parameters had remained free, you would of course get a better approximation (however that is measured), but if you are approximating something which has continuous derivatives, the constraints may very well reduce the approxmation error relative to the number of free parameters (computational cost).

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