This leads to Hermite interpolation problems.

Two-point G2 Hermite interpolation with spirals by inversion of hyperbola.

Confluent Vandermonde matrices are used in Hermite interpolation.

The general Hermite interpolation problem asks for a polynomial taking the prescribed derivatives in each node :

These were put together in 2000-frame animations that were then motion-captured; movement from one animation to another was blended out with the use of Hermite interpolation.

In numerical analysis, Hermite interpolation, named after Charles Hermite, is a method of interpolating data points as a polynomial function.

Unlike Newton interpolation, Hermite interpolation matches an unknown function both in observed value, and the observed value of its first m derivatives.

In contrast to Lagrange interpolation and Hermite interpolation, a Birkhoff interpolation problem does not always have a unique solution.

Hermite interpolation problems are those where not only the values of the polynomial p at the nodes are given, but also all derivatives up to a given order.

It differs from Hermite interpolation in that it is possible to specify derivatives of p at some points without specifying the lower derivatives or the polynomial itself.

Hermite polynomials, Hermite interpolation, Hermite normal form, Hermitian operators, and cubic Hermite splines are named in his honor.

This prefix based approach can also be used to obtain the generalized divided differences for (confluent) Hermite interpolation as well as for parallel algorithms for Vandermonde systems.

As pointed out in MSDN and OpenGL documentation, smoothstep implements cubic Hermite interpolation after doing a clamp: