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Second order Barycentric interpolation

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I already know that if I have a triangle with vertices v0, v1, v2 in property p0, p1, p2  then the property linearly interpolated at location l = a*p+ b*p1+(1-a-b)*p2

a=(area of triangle formed by the points v1, vand l)  / (area of the triangle formed by points v0, v1, v2)  

b= (area of triangle formed by the points v0, v2, and l) / (area of the triangle formed by points v0, v1, v2)  

My question is how do I generalize this to a higher order interpolation method? In my problem the property p = the normal vector to a surface and I have already calculated the Gaussian curvature of the surface to decide whether to subdivide the triangle into more triangles, but I don't want the second-order partial derivatives to go to waste.  

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Instead of interpolating the normal factors I could interpolate a function of the normal vectors such as diffuse and specular lighting which is still a function of second-order partial derivatives but the question is still the same.  If I have a property that is a function of the u, v parameters of a surface, and I also have the rate of change of the function I am trying to interpolate then how do I factor in the rate of change at the vertices of a triangle as well as the properties of the triangle at the vertices?

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