B₂-splines

B₂-splajn is similar to B-spline. Instead of 2m+1 B-spline control vertexes $D_{0},\ldots,D_{2m}$, B₂-spline is determined by odd control vertexes $D_{0}, D_{2},\ldots, D_{2m}$ and even control points are replaced by points $P_{i}, \ i=1,\ldots,m$, joining segments Q_2i-1 and Q_2i. We will consider special case of B₂ -splines, when knot set is equidistant, in other words knot u_i=i. In this case point P_i holds an equation:

$$P_{i}=\frac{1}{6}(D_{2i-2}+4D_{2i-1}+D_{2i})$$ (2)

To construct blending functions we need B₂-spline functions. It is easy to derive vertices of B₂-spline function: D_2i=(i, d_2i), P_i=(i,p_i). We do not need to write first coordinate, so we will consider each control point to be identical to its second coordinate. Next equations show, how we can count points of segments Q_2i-1 and Q_2i:

$$Q_{2i-1}(t)=(t^{3},t^{2},t,1) \frac{1}{24} \left( \begin{array}{r... ...array}{l} D_{2i-2}\\ P_{i}\\ D_{2i}\\ P_{i+1}\\ D_{2i+2} \end{array}\right)$$ (3)

$$Q_{2i}(t)=(t^{3},t^{2},t,1)\frac{1}{24} \left( \begin{array}{rrrr... ...array}{l} D_{2i-2}\\ P_{i}\\ D_{2i}\\ P_{i+1}\\ D_{2i+2} \end{array}\right)$$ (4)

where t=u-(2i-1) resp. t=u-2i.