Recursive Method
pros : Easier to implement; No need for extra loops
cons : Need extra memory for function calls
Non-recursive Method
pros : Faster than recursive method; No need for ask for extra memory for function calls
cons : Need an array and need to use for-loop
Files : main.m(for testing) , BlendingU.m(recursive) ,
BlendingFun.m(non-recursive)
In the following, I will show how to obtain Np3 (p=1,2,3, ..., 8) and get the plot of the Blending function which will be shown in the end. Note one can change the 3 to 1 and 2 to see the difference. I mean to obtain Np1 and Np2.
main.m
clear
clc
%% Knot vector & Normalization
U=[0 0 0 1 2 3 4 4 5 5 5];
U=U/max(U);
%% Non-recursive
for u=0:0.0001:1
for p=1:1:8
N=BlendingFun(p,3,u,U);
subplot(211),plot(u,N);hold on
end
end
hold off
%% Recursive
for u=0:0.0001:1
for p=1:1:8
N=BlendingU(p,3,u,U);
subplot(212),plot(u,N);hold on
end
end
BlendingU.m
%% Recursive method to compute the blending function
%% Pros : Easier to implement; No need for extra loops
%% Cons : Need extra memory for function calls
function N=BlendingU(n1,pp,uu,Ubar)
if pp==1
if uu>Ubar(1,n1) & uu<Ubar(1,n1+1)
N=1;
else
N=0;
end
else
N1N=BlendingU(n1,pp-1,uu,Ubar);
N2N=BlendingU(n1+1,pp-1,uu,Ubar);
N=0;
if N1N~=0 & Ubar(1,n1+pp-1)-Ubar(1,n1)~=0
N1=(uu-Ubar(1,n1))/(Ubar(1,n1+pp-1)-Ubar(1,n1));
N=N1*N1N;
end
if N2N~=0 & Ubar(1,n1+pp)-Ubar(1,n1+1)~=0
N2=(Ubar(n1+pp)-uu)/(Ubar(1,n1+pp)-Ubar(1,n1+1));
N=N+N2*N2N;
end
end
BlendingFun.m
%% None-recursive method to compute the blending function
%% Only the associated k elements in U are used in this function
%% Need a 1 by (1+k)*k*0.5 array, say ARRAY in this function
%% Pros : Faster than recursive method; No need for ask for extra memory for
%% function calls
%% Cons : Need an array and need to use for-loop
function N=BlendingFun(i,k,u,U)
U=U(i:i+k);
ARRAY=zeros(1,(1+k)*k*0.5);
for ind=0:1:k-1
if u>=U(ind+1) & u<U(ind+1+1)
ARRAY(ind+1)=1;
end
end
layer=2;
pt=k+1;
while layer<=k
for x=0:1:(k+1)-layer-1
if U(x+1+layer-1)-U(x+1)~=0 & ARRAY(pt+x-(k+1-layer)-1)~=0
ARRAY(pt+x)=((u-U(x+1))/(U(x+1+layer-1)-U(x+1)))*ARRAY(pt+x-(k+1-layer)-1);
end
if U(x+1+1+layer-1)-U(x+1+1)~=0 & ARRAY(pt+x-(k+1-layer))~=0
ARRAY(pt+x)=ARRAY(pt+x)+((U(x+1+1+layer-1)-u)/(U(x+1+1+layer-1)-U(x+1+1)))*ARRAY(pt+x-(k+1-layer));
end
if x==(k+1)-layer-1
layer=layer+1;
pt=pt+(k+2)-layer;
end
end
end
%% Return the last element of ARRAY
N=ARRAY((1+k)*k*0.5);
