I have a large list of data points (x,y,z). I have plotted them using ListPlot3D and then i generated a smooth surface covering the points using InterpolationOrder->3. Now i need to find the volume fraction under this smooth surface. How can i do this, so the processing is fast.

I have tried BoundaryDiscretizeGraphics@Show@DiscretizeGraphics (Volume under a List3dPlot?) but the evaluation does not complete even after a couple of hours. Any help would be greatly appreciated.

  • $\begingroup$ If the x,y coordinates form a grid consider using Interpolation and NIntegrate. I'm not sure visualizing something is the best way to integrate it. $\endgroup$
    – user1722
    Aug 29, 2018 at 16:27
  • 1
    $\begingroup$ It would help, if you would provide more detailes about what you have already tried. $\endgroup$
    – Johu
    Aug 29, 2018 at 18:48

1 Answer 1


Your data points lie on a regular grid, right? Than you can apply the trapezoidal rule directly to the data set:

First, I have to generate a fictive data set: pts are the coordinates in the plane and z denotes the elevation of the surface (yes, I presume that your surface is actually a graph).

xmin = 0.; xmax = 1.; xn = 1000;
ymin = 0.; ymax = 1.; yn = 1000;
pts = Tuples[{Subdivide[xmin, xmax, xn], Subdivide[ymin, ymax, yn]}];
f = {x, y} \[Function] x^2 x Sin[5 x + 3 y] + 1/2 Sin[7 x + 13 Pi y];
z = f @@ Transpose[pts];

This computes the integral with the two-dimensional trapezoidal rule (utilizing the fact that the data lies on a tensor product grid). If your surface is of class $C^2$, then the error should be proportional to the square of the diagional of the grid quadrilaterals.

xω = ConstantArray[1., xn + 1]; xω[[1]] = 0.5; 
xω[[-1]] = 0.5;
yω = ConstantArray[1., yn + 1]; yω[[1]] = 0.5; 
yω[[-1]] = 0.5;
int1 = (xmax - xmin)/xn (ymax - ymin)/yn (yω.Partition[z, xn + 1].xω);


For checking the accuracy, here is the same integral computed with NIntegrate:

int2 = NIntegrate[ f[x, y], {x, xmin, xmax}, {y, xmin, xmax}, AccuracyGoal -> 10, Method -> "GaussKronrodRule" ]


There is definitely some error, but for inaccurate input data, this is really negligble:

Abs[int1 - int2]/Abs[int2]


General hint

Make sure that your data z = data[[All,3]] is a packed array of machine precision numbers, for example with z = Deverloper`ToPackedArray[data[[All,3]]].

  • $\begingroup$ Thank you this helped with the problem. But i managed to use a completely different method to solve this problem $\endgroup$ Sep 5, 2018 at 15:43
  • $\begingroup$ You're welcome. If you found an approach that suited your needs even better, why not sharing it with the community and self-answer your question? $\endgroup$ Sep 5, 2018 at 17:24
  • $\begingroup$ i just needed the volume fraction, so i summed all the z-values in relation to the grid ( since the x and y values are similar to that of the grid). So i didn't have to perform a complex integration. $\endgroup$ Sep 5, 2018 at 18:51

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.