The B-Spline Approximation

If (n+1) ordered position vectors Q₀, Q₁, ..., Q_n−1, Q _n are given (Fig. 6.1), consider the (n−2) linear combinations: 6.1$$ {P_i}(t) = {X_0}(t){Q_{i - 1}} + {X_1}(t){Q_i} + {X_2}(t){Q_{i + 1}} + {X_3}(t){Q_{i + 2}}(i = 1,2,...,n - 2)$$ each formed from four successive points. X₀(t), X₁(t), X₂(t) and X₃(t) are polynomials in the parameter t(0≦t≦1). P _i (t) is a curve segment expressed in terms of the varying parameter. The condition for two neighboring curve segments P _i (t) and P_i+1(t) to be continuous at the point corresponding to t=1 for the first segment and t=0 for the second, that is, for P _i (1)= P_i+1(0) to hold for all Q _i (j=i−1, i, ..., i+3), is: 6.2$$ \left. {\begin{array}{*{20}{l}} {{X_0}(1) = {X_3}(0) = 0} \\ {{X_1}(1) = {X_0}(0)} \\ {{X_2}(1) = {X_1}(0)} \\ {{X_3}(1) = {X_2}(0)} \end{array}} \right\} ]$$Fig. 6.1Derivation of a B-spline curve (case of M=4)