%  Construct a circle
t = linspace(0,4);
x = cos(pi*t/2);
y = sin(pi*t/2);
T = [0 1 2 3 4];
X = cos(pi*T/2);
Y = sin(pi*T/2);
plot(t,x,t,y,T,X,'o',T,Y,'o'), axis('equal'),grid on
pause

plot(x,y,X,Y,'o'), axis('equal'), grid on
pause

% polynomial interpolation
a = polyfit(T,X,4);
b = polyfit(T,Y,4);
p = polyval(a,t);
q = polyval(b,t);
plot(t,x,t,y,t,p,t,q,T,X,'o',T,Y,'o'), axis('equal'),grid on
pause

plot(x,y,X,Y,'o',p,q), axis('equal'), grid on
pause

% polynomial interpolation 2
T = [0.2 1 2 3 3.8];
X = cos(pi*T/2);
Y = sin(pi*T/2);
a = polyfit(T,X,4);
b = polyfit(T,Y,4);
p = polyval(a,t);
q = polyval(b,t);
plot(t,x,t,y,t,p,t,q,T,X,'o',T,Y,'o'), axis('equal'),grid on
pause

plot(x,y,X,Y,'o',p,q), axis('equal'), grid on
pause

% Hermite 
T = [0 1 2 3 4];
X = cos(pi*T/2);
Y = sin(pi*T/2);
SX = -pi*Y/2;
SY = pi*X/2;
a = pwch(T,X,SX);
b = pwch(T,Y,SY);
p = ppval(a,t);
q = ppval(b,t);
plot(t,x,t,y,t,p,t,q,T,X,'o',T,Y,'o'), axis('equal'),grid on
pause

plot(x,y,X,Y,'o',p,q), axis('equal'), grid on
pause

% Not a knot
a = spline(T,X);
b = spline(T,Y);
p = ppval(a,t);
q = ppval(b,t);
plot(t,x,t,y,t,p,t,q,T,X,'o',T,Y,'o'), axis('equal'),grid on
pause

plot(x,y,X,Y,'o',p,q), axis('equal'), grid on
pause

% clamped spline
SX = [ 0 X 0];
SY = [pi/2 Y pi/2];
a = spline(T,SX);
b = spline(T,SY);
p = ppval(a,t);
q = ppval(b,t);
plot(t,x,t,y,t,p,t,q,T,X,'o',T,Y,'o'), axis('equal'),grid on
pause

plot(x,y,X,Y,'o',p,q), axis('equal'), grid on
pause

% shape preserver
a = pchip(T,X);
b = pchip(T,Y);
p = ppval(a,t);
q = ppval(b,t);
plot(t,x,t,y,t,p,t,q,T,X,'o',T,Y,'o'), axis('equal'),grid on
pause

plot(x,y,X,Y,'o',p,q), axis('equal'), grid on