Cubic Hermite spline

Remarks

The piece on $[x_i, x_{i+1}]$ is the unique cubic with $s(x_i) = p_i$, $s'(x_i) = m_i$, $s(x_{i+1}) = p_{i+1}$ and $s'(x_{i+1}) = m_{i+1}$, that is, a Hermite interpolant. Check: $h_{00}(0) = 1$, $h_{00}(1) = 0$, $h_{01}(1) = 1$, $h_{10}'(0) = 1$, $h_{11}'(1) = 1$, and the other values and slopes at $0$ and $1$ vanish. So $s$ is a spline of class $C^1$, but in general not of class $C^2$. Example: nodes $0, 1$ with $p = (0, 1)$ and $m = (0, 0)$ give $s(x) = 3x^2 - 2x^3$. Non-example: the piecewise linear interpolant of the same values is not a cubic Hermite spline for these slopes, since its slope on $[0, 1]$ is $1$, not $0$. When the slopes are not given, they are chosen from the values, for instance by finite differences (Catmull–Rom spline) or by monotonicity-preserving rules.