how to make a smooth spline connection at endpoints?

I can draw a smooth spline across points but when I try to join the end to the beginning I get a sharp corner instead of a smooth connection. How can I smooth out the endpoint connection to make a smoothly shaped blob? Here's my code:

x1 = [0 2 1 4 3 2]; 
y1 = [1 0 2 1 3 2];
tnum = 1000;
t = linspace(0,1,tnum); 
x1 = [x1 x1(1)]; % ADDING THE FIRST POINT TO THE END
y1 = [y1 y1(1)]; % ADDING THE FIRST POINT TO THE END
[xt yt] = ParametricSpline(x1, y1); 
xref = ppval(xt, t);
yref = ppval(yt, t); 
plot(x1, y1, 'o', xref, yref);

function [xtF, ytF] = ParametricSpline(x,y) 
    arc_length = 0; 
    n = length(x); 
    t = zeros(n, 1);
    for i=2:n 
        arc_length = sqrt((x(i)-x(i-1))^2 + (y(i)-y(i-1))^2); 
        t(i) = t(i-1) + arc_length; 
    end
    t=t./t(length(t)); 
    xtF = spline(t, x); 
    ytF = spline(t, y);
end

[darova]

What happens if you do spline twice (add the same points)?

[mark palmer]

I thought of that too last night, will try, but I suspect there should be a more elegant solution.

Answer 1

[Ameer Hamza]

Try csape:

x1 = [0 2 1 4 3 2]; 
y1 = [1 0 2 1 3 2];
tnum = 1000;
t = linspace(0,1,tnum); 
x1 = [x1 x1(1)]; % ADDING THE FIRST POINT TO THE END
y1 = [y1 y1(1)]; % ADDING THE FIRST POINT TO THE END
[xt yt] = ParametricSpline(x1, y1); 
xref = ppval(xt, t);
yref = ppval(yt, t); 
plot(x1, y1, 'o', xref, yref);
function [xtF, ytF] = ParametricSpline(x,y) 
    arc_length = 0; 
    n = length(x); 
    t = zeros(n, 1);
    for i=2:n 
        arc_length = sqrt((x(i)-x(i-1))^2 + (y(i)-y(i-1))^2); 
        t(i) = t(i-1) + arc_length; 
    end
    t=t./t(length(t)); 
    xtF = csape(t, x, 'perodic'); 
    ytF = csape(t, y, 'perodic');
end

Result:

[Akira Agata]

Or you can use spline function, like:

x1 = [0 2 1 4 3 2]; 
y1 = [1 0 2 1 3 2];
x1 = [x1 x1(1)]; % ADDING THE FIRST POINT TO THE END
y1 = [y1 y1(1)]; % ADDING THE FIRST POINT TO THE END
t = linspace(0,2*pi,numel(x1));

pp = spline(t,[0 x1 0; -1 y1 -1]); % Assuming an endslope at (0,1) is (0, -1)
xy2 = ppval(pp, linspace(0,2*pi));

figure
scatter(x1,y1)
hold on
plot(xy2(1,:),xy2(2,:))