I am trying to generalize the principle of Hermite interpolation to 2D, over irregular data points.
In 1D, we have a sequence of points on which the value of a function and its first derivative is known. The Hermite interpolant is a piecewise cubic function such that on every interval a cubic polynomial is defined by the two values and two derivatives at the endpoints. This is enough to ensure $C_0$ and $C_1$ continuity.
Now in 2D, we can generalize by defining piecewise polynomial functions over a triangulation of the points, where every triangular patch is just defined by the function values and gradient values at the vertices. To ensure $C_0$ and $C_1$ continuity ("tangent plane continuity"), I want to enforce the following properties:
The interpolant over a triangle is a bivariate polynomial defined by the values and gradient component values at the vertices ($9$ numbers);
Furthermore, the values and gradient values along the edges are only depending on the (gradient) values at the two vertices they link.
I am looking for a simple way to establish suitable formulas, with a polynomial degree as low as possible. Notice that if we follow en edge, the gradient in the direction of the edge can be obtained as an ordinary $1D$ Hermite interpolation; but for the gradient in the transverse direction, I have no precise clue. I assume that the barycentric coordinates can be helpful here.
Is this kind of interpolant already known ? How can I address the problem ?
Note that I am not after a method to compute the gradients at vertices, just a way to enforce the continuity condition along edges.