# How to calculate the field on the surface created with Delaunay triangulation?

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Nima on 18 Oct 2020
Commented: Nima on 18 Oct 2020
In the following code x, y, z corresponde to the point coordinates in a volume, and then a surface is created using Delaunay triangulation.
The values of magnetic flux density at the points are B_x, B_y, B_z . The problem is to estimate or calculate (eg. interpolation or ...) the 3D magnetic fields on this surface. I will appriciate of any idea.
x = [-0.76; -1.82; -0.81; -0.35; 0.05; -0.23; -0.72; -0.85; 0.87; 0.73; -1.86; 1.50; -0.90; -1.63; 0.31; 0.01; -0.04; -1.92; -0.61; 0.83];
y = [0.04; 0.65; -1.07; -1.34; -1.70; 1.40; -1.53; -1.25; 1.42; 0.67; 0.01; -0.38; 1.42; -0.96; 1.92; -1.86; 1.46; -0.07; -1.84; -0.68];
z = [1.79; 0.18; -1.40; -1.36; -0.94; 1.32; -0.95; 1.22; -1.00; 1.67; -0.58; -1.18; -0.97; 0.45; -0.10; -0.57; 1.29; 0.33; 0.19; 1.62];
B_x = [0.00014; -0.00014; 0.0002; -0.00019; -0.00024; 0.00025; -0.00023; 0.00016; -0.00016; 4.62e-05; 1.56e-05; 8.07e-05; 0.00013; 0.00017; -0.00025; 0.00025; -0.00026; -0.00017; 0.00017; 6.24e-05];
B_y = [1.24e-09; 2.92e-08; -1.30e-07; 7.09e-08; -2.02e-07; -1.23e-07; -1.62e-07; 4.57e-08; 1.52e-07; 1.71e-07; 7.88e-08; -2.14e-07; -1.38e-07; -1.14e-07; 2.63e-07; 1.31e-07; -1.09e-07; -5.48e-09; -1.36e-07; 7.82e-08];
B_z = [-0.0001; -0.0001; -7.51e-05; -8.704e-05; -1.76e-05; 0.0001; 0.00017; 0.00028; 0.0003; 0.0003; 0.0003; 0.0003; 0.00029; 0.00027; 0.00014; 0.00014; 6.78e-05; -0.0001; -0.0001; -0.00016];
T = delaunayTriangulation(x,y,z) % Delaunay Triangulation
figure(1)
plot3(x,y,z,'ro')
figure(2)
[K,v] = convexHull(T);
trisurf(K,T.Points(:,1),T.Points(:,2),T.Points(:,3))

Bjorn Gustavsson on 18 Oct 2020
If you have the magnetic field components at the points [x,y,z] then you should be able to use TriScatteredInterp or (preferably, since it is "newer") scatteredInterpolant. Either for the individual components, or for the magnetic field-strength to re-interpolate to whatever grid you want. For one component it would look something like this:
f_Bx = scatteredInterpolant([x(:),y(:),z(:)],B_x,'natural');
Bx_i = f_Bx([xi(:),yi(:),zi(:)]);% Here xi etc are your re-interpolation-grid-points
If your surface is reasonably flat you might get a neater result if you exclude the variable with the small variation.
HTH
Nima on 18 Oct 2020
I want the field on the surface figure(2) not inside the volume figure(3)...my question is: what is the appropriate grid for interpolating on this surface?

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