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Next: Using blending functions to Up: SPLINE - BLENDED SURFACES Previous: B2-splines

Construction of blending functions

In this section we will construct blending functions Lmi(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 Lmi(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:

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

This is equivalent to conditions:

\begin{displaymath}
P_{i}^{i}=1,\ P_{k}^{i}=0,\ i\neq k,\ i=1,\ldots,m
\end{displaymath} (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:

\begin{displaymath}
\sum_{i=1}^{m} L_{i}^{m}(u)=1\ \forall u
\end{displaymath} (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):

\begin{displaymath}
\sum_{i=1}^{m} P^{i}_{k}=1 \ \ \ \
\sum_{i=1}^{m} D^{i}_{2k}=1
\end{displaymath} (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 Pii and knot 2i-1. The optimal number of non-zero segments appears to be 12. We can see from equations (3), (4), that segments Qi2j-1 and Qi2j are controled by vertexes Di2j-2,Pij,Di2j,Pij+1, Di2j+2. If any of this two segments is identically equal to zero and Pij=Pij+1 =0, then also Di2j-2=Di2j=Di2j+2=0. If we have just 12 non-zero segments, then just four odd control vertexes Di2i-4,Di2i-2,Di2i, Di2i+2 are non-zero. Obviously the only non-zero joining vertex is Pii=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{displaymath}
\begin{array}{ccccccccccccc}
D_{0} & P_{1} & \ldots & D_{...
... & 1 & \frac{1}{2}+p &
0 & -p & \ldots & 0 & 0
\end{array}
\end{displaymath}

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 Lm-1, Lm0, Lmm+1 and Lmm+2. In definition of Gordon surface we can use for example Lm-1+Lm0+Lm1 instead of Lm1 and Lmm+Lmm+1+Lmm+2 instead of Lmm.
next up previous
Next: Using blending functions to Up: SPLINE - BLENDED SURFACES Previous: B2-splines

1999-04-09