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HermiteSpline

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.

hermite spline

Cubic Hermite Spline

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

define cubic spline
deltax

These are the constraints that must be met.

formularise cubic spline

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

matrix cubic spline
coef cubic spline
hermite basic cubic

Quintic Hermite Spline

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.

define quintic spline
formularise quintic spline

coef quintic spline
hermite basic quintic

Higher-Order Hermite Spline

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

Septic

coef septic spline
hermite basic septic
hermite basic septic zoom

Nonic

coef nonic spline