6.2 Definition A parametric non-uniform B-spline curve is defined by P(u)=∑PMk(u) where, Pi are n+ 1 control points; Nik(u) are piecewise polynomial B-spline basis functions of order k (or degree k-1)with n> k-1 Therefore n, k are independent, unlike Bezier curves. The parameter u obeys the inequality u .2 6.2.1 Knot vector For open(non-periodic)curves, it is usual to define a set T of non-decreasing real numbers which is called the knot vector. as follows T=化=三“<女≤…≤<如+1元:=t 3 k equal values At each knot value, the curve P(u) has some degree of discontinuity in its derivatives above a certain order as we will see later. The total number of knots is n +k+ l which equals the number of control points in the curve plus the curve's order 6.2.2 Properties and definition of basis function Nik(u) 1. 2i_o Ni k(u)=l(partition of unity 3. Nik(u)=0 if u ti, ti+k(local 4. Ni k(a)is(k-2) times continuously differentiable at simple knots If a knot t, is of multiplicity p(< k), ie. if then Nik(u) is(k-p-1)times continuously differentiable, ie. it is of class CK-p-6.2 Definition A parametric non-uniform B-spline curve is defined by P(u) = Xn i=0 PiNi,k(u) (6.1) where, Pi are n + 1 control points; Ni,k(u) are piecewise polynomial B-spline basis functions of order k (or degree k − 1) with n ≥ k − 1. Therefore n, k are independent, unlike B´ezier curves. The parameter u obeys the inequality to ≤ u ≤ tn+k (6.2) 6.2.1 Knot vector For open (non-periodic) curves, it is usual to define a set T of non-decreasing real numbers which is called the knot vector, as follows: T = {to = t1 = · · · = tk−1 | {z } k equal values < tk ≤ tk+1 ≤ · · · ≤ tn | {z } n−k+1 internal knots < tn+1 = · · · = tn+k | {z } k equal values } (6.3) At each knot value, the curve P(u) has some degree of discontinuity in its derivatives above a certain order as we will see later. The total number of knots is n + k + 1 which equals the number of control points in the curve plus the curve’s order. 6.2.2 Properties and definition of basis function Ni,k(u) 1. Pn i=0 Ni,k(u) = 1 (partition of unity). 2. Ni,k(u) ≥ 0 (positivity). 3. Ni,k(u) = 0 if u 6∈ [ti ,ti+k] (local support). 4. Ni,k(u) is (k − 2) times continuously differentiable at simple knots. If a knot tj is of multiplicity ρ(≤ k), ie. if tj = tj+1 = · · · = tj+ρ−1 (6.4) then Ni,k(u) is (k − ρ − 1) times continuously differentiable, ie. it is of class C k−ρ−1 . 3