If you know the values of a function at the three points of a triangle, how can I determine the values at points in the interior? {details}?

Noticing the fact that on the first two triangles in the image that points of equal values lay on parallel lines, I wonder if this is the case for the larger triangle. If it is, then the parallel lines are not parallel to the sides of the triangle, and I wonder how to find the equations for the lines.

It seems to me that you are working with barycentric coordinates. Given 3 points A, B and C with corresponding values of v1, v2 and v3, any point in the triangle can be expressed as pA + qB + rC, where p+q+r =1 and p>0, q>0 and r >0. The associated value at the point is pv1 + qv2 +rv3. I don’t know of any way of converting from Cartesian to barycentric coordinates other than setting up the obvious 3 linear equations and solving for p, q and r. Check here I just looked at it briefly. It seems that the problem can be reduced to solving two equations.

The equation is a simple linear equation in two variable. F(x,y) = ax +by +c
You have three boundary values (the three points on the triangle.)
Substitute the values at the ponts:
F(P1)=a(P1.X)+b(P1.Y)+c
F(P2)=a(P2.X)+b(P2.Y)+c
F(P3)=a(P3.X)+b(P3.Y)+c
Solve for a,b, and c.
Now you have an equation that will take any x and y and give you F(x,y)
F(x,y) = ax +by +c

I think it has to be a plane. 3 non-co-linear points define a plane so solve for the plane and you got it.

Triangle impies plane.