Help with implementing parametric spline interpolation

Hi.

In the attachments I have a matlab-file where I am seeking a quadratic interpolant of type l_i(t) = c_i * t^2 + b_i * t + a_i.

Can you help me finding a solution for a quadratic interpolant of type l_i(t) = a_i*t^3+b_i*t^2+c_i*t+d_i ?

This is what I've done so far:

% 2aa., per coordinate and per interpolant: 4 equations:

% (1) l_i(0) = d_i = p_i

% (2) l_i(1) = a_i + b_i + c_i + d_i = p_{i+1}

% (3) l_i'(1) = l{i+1)'(0)

% --> 3 * a_i + 2 * b_i + c_i - c_{i+1} = 0

% (4) l_i''(1) = 6 * a_i + 2 * b_i - b_{i+1} = 0

e = 4 %% number of equations per interpolant and per coordinate

A = zeros(e*n); %% empty matrix-setup

bX = zeros(e*n, 1); %% empty setup of the right-hand side (for x)

bY = zeros(e*n, 1); %% empty setup of the right-hand side (for y)

for i = 0:n-1

% first equation per interpolant and per coordinate

A(1+e*i, 1+e*i) = 1;

bX(1+e*i) = p(1+i, 1);

bY(1+e*i) = p(1+i, 2);

% second equation per interpolant and per coordinate

A(2+e*i, 1+e*i) = 1;

A(2+e*i, 2+e*i) = 1;

A(2+e*i, 3+e*i) = 1;

%A(2+e*i, 4+e*i) = 1;

bX(2+e*i) = p(2+i, 1);

bY(2+e*i) = p(2+i, 2);

% third equation per interpolant and per coordinate

if i < n-1

A(3+e*i, 2+e*i) = 1;

A(3+e*i, 3+e*i) = 2;

A(3+e*i, 5+e*i) = -1;

%A(3+e*i, 7+e*i) = -2;

else

A(e*n, 2) = 1;

bX(e*n) = 1;

bY(e*n) = 1;

end

%fourth equation per interpolant and per coordinate

if i < n-2

A(4+e*i, 3+e*i) = 1;

A(4+e*i, 4+e*i) = 2;

A(4+e*i, 7+e*i) = -1;

%A(4+e*i, 7+e*i) = -2;

else

A(e*n, 2) = 1;

bX(e*n) = 1;

bY(e*n) = 1;

end

end

Best regards,

Eirik