Finding a vector field from a set of points

Question

I have a data set of the type $\{x,y,z\}$ where $(x,y)$ is a point and $z$ is the "value" or magnitude at that point. This gives me a triangular sort of shape since $(x,y,z)$ is not defined along the whole possibility set considering it is real data. Is there any way I can get a gradient using the points I have (using them if possible) and using the interpolation from Mathematica otherwise?

Answer 1

I'll try to give you an answer complementary to Szabolcs’, because I understood your question in another way. If by “get a gradient using the points I have” you mean having a continuous color gradient in your plot of the data (and not calculating a gradient, as in “derivative”), then you can simply use DensityPlot with an Interpolation.

Let's get some data, taken at random points from the function $z=\sin x\ \sin y$:

points = RandomReal[{-5, 5}, {500, 2}];
data = {#1, #2, Sin[#1]*Sin[#2]} & @@@ points;

then plot it:

DensityPlot[
 Interpolation[data, InterpolationOrder -> 1][x, y], {x, -5, 
  5}, {y, -5, 5}, PlotRange -> {Full, Full, {-1, 1}}]

which you can compare to the original function I drew points from:

DensityPlot[Sin[x]*Sin[y], {x, -5, 5}, {y, -5, 5}]

Answer 2

Your question is not very clearly formulated, but if I understand it correctly:

• you have the value of a function in a set of points on the plane

• the points do not form a rectangular lattice

• you need to estimate the gradient of the function

You can do this by interpolating linearly, then taking the gradient of the interpolated function.

Example:

Let's generate some data:

points = RandomReal[{-2, 2}, {20, 2}];
data = {points, Exp[-#.# & /@ points]}\[Transpose];

Let's interpolate:

if = Interpolation[data, InterpolationOrder -> 1]

You can get the gradient by

D[if[x, y], {{x, y}}]

The interpolated function looks like this:

ListPlot3D[Flatten /@ data, Mesh -> All]

(Note: here I used ListPlot3D, which does the interpolation internally. It uses the same method as Interpolation. I did this because it made it easy to only plot the function inside the convex hull of the points, and not extrapolate outside of that.)