Construction of blending functions

In this section we will construct blending functions L^m_i(u) as B₂-splajn. Let $P^{i}_{1},\ldots,P^{i}_{m}$ be joining points and let $D^{i}_{0},D^{i}_{2},\ldots,D^{i}_{2m}$ be odd control vertexes of blending function L^m_i(u). We have to change equation (1), because knot set is $-1, 0, 1, \ldots,2m+1$ and one set of curves contains just m curves:

$$L_{i}^{m}(2i-1)=1,\ L_{i}^{m}(2k-1)=0, i\neq k,\ i=1,\ldots,m$$

This is equivalent to conditions:

$$P_{i}^{i}=1,\ P_{k}^{i}=0,\ i\neq k,\ i=1,\ldots,m$$ (5)

Conditions (1) determine joining points $P^{i}_{1},\ldots,P^{i}_{m}$. Odd control vertexes do not need to satisfy any conditions. We have to find appropriate odd control vertexes. Firstly we want the points of surface G₁(u,v) to be a barycentric combination of points of curves $G(u_{i},v), i=1,\ldots,m$. This is satisfied, if the next equation is satisfied:

$$\sum_{i=1}^{m} L_{i}^{m}(u)=1\ \forall u$$ (6)

We can thing about surfaces G₂(u,v) and G₁₂(u,v) in the same way. We can easy get this conditions from equations (3), (4) and (6):

$$\sum_{i=1}^{m} P^{i}_{k}=1 \ \ \ \ \sum_{i=1}^{m} D^{i}_{2k}=1$$ (7)

Secondly, we want the given curves to influent the surface just locally. Therefore we want blending function to be non-zero just over few segments around point Pⁱ_i and knot 2i-1. The optimal number of non-zero segments appears to be 12. We can see from equations (3), (4), that segments Qⁱ_2j-1 and Qⁱ_2j are controled by vertexes Dⁱ_2j-2,Pⁱ_j,Dⁱ_2j,Pⁱ_j+1, Dⁱ_2j+2. If any of this two segments is identically equal to zero and Pⁱ_j=Pⁱ_j+1 =0, then also Dⁱ_2j-2=Dⁱ_2j=Dⁱ_2j+2=0. If we have just 12 non-zero segments, then just four odd control vertexes Dⁱ_2i-4,Dⁱ_2i-2,Dⁱ_2i, Dⁱ_2i+2 are non-zero. Obviously the only non-zero joining vertex is Pⁱ_i=1. Thirdly we want each given curve to have the same influence to Gordon surface. This will be satisfied, if $D^{i}_{2i-4}=a, D^{i}_{2i-2}=b ,D^{i}_{2i}=c, D^{i}_{2i+2}=d, \forall i$, where a,b,c,d are real numbers. Finally we want each blending function to be symmetrical. That means a=d and b=c. We did not satisfy conditions (7) yet. First one is obviously satisfied. Now we can replace second one by new simplier one: 2a+2b=1. Let p be a real parameter. We will set a=-p and $b=p+\frac{1}{2}$. This is the result of construction. Now we can assume all verexes of blending functions:

$$\begin{array}{ccccccccccccc} D_{0} & P_{1} & \ldots & D_{... ... & 1 & \frac{1}{2}+p & 0 & -p & \ldots & 0 & 0 \end{array}$$

Now we need to note, that condition (6) for odd vertexes is not satisfied for $k \in \{0,1,m-1,m\}$. We need to increase the number of blending functions. We need also L^m_-1, L^m₀, L^m_m+1 and L^m_m+2. In definition of Gordon surface we can use for example L^m_-1+L^m₀+L^m₁ instead of L^m₁ and L^m_m+L^m_m+1+L^m_m+2 instead of L^m_m.