Hermite interpolation

Hermite interpolation : cubic curve interpolating points and derivatives at extremities

Consider the following constraints
\(\left\{\begin{array}{l} p(s) = c_3 s^3 + c_2 s^2 +c_1 s +c_0 \\ p(0)=p_0, \; p(1)=p_1, \;p'(0)=d_0, \;p'(1)=d_1\end{array}\right.\)

\(\Rightarrow\) System of equations
\(\left\{\begin{array}{rrrr} &&&c_0& = p_0 \\ c_3\,+&c_2\,+&c_1\,+&c_0& = p_1 \\ &&c_1\;\;\;\,&& = d_0 \\ 3\, c_3\, +& 2\, c_2\, + &c_1\,\;\;\;&& = d_1 \\ \end{array} \right.\) \(\Rightarrow\) \(\left\{\begin{array}{llll} c_0=p_0 \\ c_1=d_0 \\ c_2 = -3 p_0 +3 p_1 -2 d_0 - d_1 \\ c_3 = 2 p_0 -2 p_1 + d_0 + d_1 \end{array}\right.\)

• \(\forall s\in[0,1]\;,\;\; p(s) = (2s^3-3s^2+1) \; p_0 + (s^3-2s^2+s) \; d_0 + (-2s^3+3s^2) \; p_1 +(s^3-s^2) \; d_1\)

• For arbitrary \(t\in[t_i,t_{i+1}]\), we set \(\displaystyle s=\frac{t-t_i}{t_{i+1}-t_i}\), \(\tilde d_{0/1}=(t_{i+1}-t_i)\,d_{0/1}\)