Question

How would one interpolate a vector field defined in 3d? Suppose I have experimentally determined the coordinates of a finite number of vectors defined over a 3d grid, but I want to be able to extract a vector direction from any point within the boundaries of the grid.

Help is appreciated!

Answer 1

im not sure i understand what your asking, but given, say, 3 vectors (or points in x,y,z), we can devise an equation that should match them. When the elements of the first point is used, the others zero to find the "constant".
\[<x_1,y_1,z_1>;<x_2,y_2,z_2>;<x_3,y_3,z_3>\]
\[z = c_1+c_2(x-x_1)(y-y_1)+c_3(x-x_1)(y-y_1)(x-x_2)(y-y_2)\]

Answer 2

Thanks for the response, let me rephrase:
My independent variables are x, y, z. At each point (x,y,z), a vector is defined with components (c1, c2, c3). So in effect I have a3d vector field. I guess my question isnt really specific, I'm looking for a general method or resource which would allow me to interpolate these vectors if I already know the positions and components of a finite set of them.

Hope this clears it up.

Answer 3

had to refresh what some of those terms meant :)

interpolate is to find either a best fit, or exact fit, equation (or function) with the given data.

a vector field defines a vector at a given point.

what it sounds like is that you have a field of tangents to a surface or curve; and are wanting to construct a suitable function that matches those charachterisics

the following is not a very easy read for me, but it does point out alot of the inner workings i believe.

http://www.flame.org/~cdoswell/publications/Schaefer%26Doswell_79.pdf

Answer 4

Haha yeah I had found that document. I'll keep on searching for a good resource, if you want I can give you news of it here.