# Consider two quadratics that meet at a middle point
x1 = -2
y1 = 3
x2 = -1
y2 = 1
x3 = 0
y3 = 2
x4 = 3
y4 = -1
x5 = 4
y5 = 0
p1 = y1*(x-x2)*(x-x3)/((x1-x2)*(x1-x3)) + y2*(x-x1)*(x-x3)/((x2-x1)*(x2-x3)) + y3*(x-x1)*(x-x2)/((x3-x1)*(x3-x2)) # using Basis approach
p2 = y3*(x-x4)*(x-x5)/((x3-x4)*(x3-x5)) + y4*(x-x3)*(x-x5)/((x4-x3)*(x4-x5)) + y5*(x-x3)*(x-x4)/((x5-x3)*(x5-x4)) # using Basis approach
G = point([(x1,y1),(x2,y2),(x3,y3),(x4,y4),(x5,y5)],size=20)
G += plot(p1,(x,x1,x3),color='red') + plot(p2,(x,x3,x5),color='orange')
show(G)
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It would be nice if where the curves met that the slopes also met. Unfortunately, there is no more "freedom" in choosing the quadratics. Need more coefficients to pick.
# This cell in Sage mode
var('u')
H0 = (1-u)^2*(1+2*u)
H1 = (1-u)^2*u
H2 = (u-1)*u^2
H3 = (3-2*u)*u^2
G = plot(H0,(u,0,1),color='black')+plot(H1,(u,0,1),color='red')+plot(H2,(u,0,1),color='green')+plot(H3,(u,0,1),color='blue')
show(G,aspect_ratio=1)
show(H0)
show(H1)
show(H2)
show(H3)
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The rest of this is in octave mode
function y = H0(u)
y = (1-u) .^2 .* (1+2*u);
end
function y = H1(u)
y = (1-u) .^2 .*u;
end
function y = H2(u)
y = (u-1) .* u .^2;
end
function y = H3(u)
y = (3-2 .*u) .* u .^2;
end
n = 4;
x = [-1 2 4 7 13];
y = [5 4.5 2 3 0];
m = [-1 -1 -1 -1 -2];
for k = 2:n+1
xx = x(k-1):0.05:x(k);
delx = x(k)-x(k-1);
yy = H0( (xx-x(k-1))/delx )*y(k-1) + H1( (xx-x(k-1))/delx )*m(k-1)*delx + H2( (xx-x(k-1))/delx )*m(k)*delx + H3( (xx-x(k-1))/delx )*y(k);
plot(xx,yy)
hold on
end
plot(x,y,'rd')
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Here is a parametric version of above. Notice the use of sign to make the slopes match graphical reality.
function y = H0(u)
y = (1-u) .^2 .* (1+2*u);
end
function y = H1(u)
y = (1-u) .^2 .*u;
end
function y = H2(u)
y = (u-1) .* u .^2;
end
function y = H3(u)
y = (3-2 .*u) .* u .^2;
end
n = 4;
t = [0 1 2 4 7];
x = [4 2 1.5 1 4];
y = [5 4.5 4 3 0];
run = [1 1 0 0 1];
rise = [-1 1 1 -1 -2];
for k = 2:n+1
tt = t(k-1):0.05:t(k);
delt = t(k)-t(k-1);
xx = H0( (tt-t(k-1))/delt )*x(k-1) + H1( (tt-t(k-1))/delt )*run(k-1)*delt*sign(x(k)-x(k-1)) + H2( (tt-t(k-1))/delt )*run(k)*delt*sign(x(k)-x(k-1)) + H3( (tt-t(k-1))/delt )*x(k);
yy = H0( (tt-t(k-1))/delt )*y(k-1) + H1( (tt-t(k-1))/delt )*rise(k-1)*delt*sign(y(k)-y(k-1)) + H2( (tt-t(k-1))/delt )*rise(k)*delt*sign(x(k)-x(k-1)) + H3( (tt-t(k-1))/delt )*y(k);
plot(xx,yy)
hold on
end
plot(x,y,'rd')
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