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*p₀ + b*p₁+(1-a-b)*p₂

a=(area of triangle formed by the points v₁, v₂ and 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.  

Bump.  Please help poster doesn't have access to internet.

 

Tried to interpolate u and v values then . - multiplied the change in u and v by the gradient of the normal vector with respect to u and v.  But it didn't work.

Bump. 

 

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?