The Bezier-curve produced by the Bernstein basis function has limited flexibility.

• First, the number of specified polygon vertices fixes the order of the resulting polynomial which defines the curve.

• The second limiting characteristic is that the value of the blending function is nonzero for all parameter values over the entire curve.

The B-spline basis contains the Bernstein basis as the special case. The B-spline basis is non-global.

A B-spline curve is defined as a linear combination of control points Pi and B-spline basis function Ni, , k (t) given by

Where, • {Pi: i=0, 1, 2….n} are the control points • k is the order of the polynomial segments of the B-spline curve. Order k means that the curve is made up of piecewise polynomial segments of degree k - 1, • the Ni,k(t),k(t) are the “normalized B-spline blending functions”. They are described by the order k and by a non-decreasing sequence of real numbers normally called the “knot sequence”.

$t_i : i = 0,... n + K$

The Ni, k functions are described as follows −

Properties of B-spline Curve

B-spline curves have the following properties −

• The sum of the B-spline basis functions for any parameter value is 1.

• Each basis function is positive or zero for all parameter values.

• Each basis function has precisely one maximum value, except for k=1.

• The maximum order of the curve is equal to the number of vertices of defining polygon.

• The degree of B-spline polynomial is independent on the number of vertices of defining polygon.

• B-spline allows the local control over the curve surface because each vertex affects the shape of a curve only over a range of parameter values where its associated basis function is nonzero.

• The curve exhibits the variation diminishing property.

• The curve generally follows the shape of defining polygon.

• Any affine transformation can be applied to the curve by applying it to the vertices of defining polygon.

• The curve line within the convex hull of its defining polygon.