Hermite Spline is a piecewise interpolation function that determines the interpolation polynomial using the values of the support points at both ends of the interval and the derivative as constraints. Unlike the B-spline, it always passes through the support point. Depending on the rule that determines the derivative at the support point, there are different properties such as monotonicity and overshoot. It is necessary to use these rules for different purposes.

The segmented polynomial of a cubic hermite spline is defined by the following equation.

These are the constraints that must be met.

From this matrix representation, the coefficients of the hermite basis functions can be obtained.

In the same way, we can find the Hermite spline of the fifth order. To use this, we also need the second derivative at the support point.

Even higher-order Hermite splines can be defined. However, these are rarely used.