I am relatively new to Mathematica and have a question regarding ListPlot3D.

I am plotting results for the 2 dimensional heat equation for a particular point in time, so to each (x,y) in my partition of the xy-plane (with x ranging from 0 to 100, y from 0 to 200), there is point in the z direction that represents the heat as a function of x and y (and time, but the plot represents the heat distribution at a fixed point in time). This was done using the finite difference method.

I initially plotted the results using ListPointPlot3D as well as ListPlot3D (which of course "interpolates" the points). I would like to plot some level curves on the xy-plane, but in absence of z as closed-form function of x and y, I do not know how to do this with Mathematica.

Essentially, what I would like to do is intersect the ListPlot3D surface with a few planes of constant height above the xy-plane, and then project the intersections on to the xy-plane, but because I do not have an explicit function to which I can equate the plane equation to, I have to find an alternative method. Does anyone have any suggestions on this?

Alternatively, is there another feature in Mathematica of which I am unaware that will compute a best-fit function, so I can supply a number of tuples of the form {x, y, z(x,y)} and get a closed-form solution of z?

Any help would be greatly appreciated.

  • $\begingroup$ Try the option MeshFunctions -> {#3 &} in ListPlot3D[]. Look up ListContourPlot[] as well. $\endgroup$ Commented Mar 25, 2018 at 20:12
  • $\begingroup$ The ListContourPlot feature works quite well, actually. Thanks. I appreciate it. $\endgroup$
    – mrmingus
    Commented Mar 25, 2018 at 20:51

2 Answers 2


You could use SliceContourPlot3D, e.g.

With[{f = Sin[x] Cos[y]}, 
 Show[Plot3D[f, {x, -3, 3}, {y, -3, 3}, Mesh -> None, 
   PlotStyle -> Opacity[0.3]
   ], SliceContourPlot3D[
   f - z - 1, {z == -1}, {x, -3, 3}, {y, -3, 3}, {z, -1, 10}, 
   PlotLegends -> Automatic]]]

enter image description here


I have been using this simple low-level trick to create slices:

  1. First, you generate separately a 3D plot and whatever 2D plot you are interested in. If I understand your discrete problem correctly, you could do this step by extracting the relevant sublists from the data (see the online doc for Select[] if you are wondering how to do it).
  2. Next, you convert your 2D Graphic into a 3D Graphic by converting each pair of points {x,y} into a triplet {x,y,z}. For a horizontal slice, I simply use z=offset but you could use any function f:{x,y}->{x’,y’,z’}.
  3. Use Show to combine both 3D graphics.

Here is an example of the code:

f[x_, y_] = x^2 + y^2;
offset = -2;
landscape = Plot3D[f[x, y], {x, -1, 1}, {y, -1, 1}, 
                   PlotStyle -> Opacity[0.7],
                   ColorFunction -> "Rainbow"];
map = ContourPlot[f[x, y], {x, -1, 1}, {y, -1, 1}];
map3D = Graphics3D[map[[1]] /. {x : _Real, y : _Real} -> {x, y, offset}];
Show[landscape, map3D, 
     PlotRange -> {Automatic, Automatic, {offset, 2}}

enter image description here

  • $\begingroup$ Elegant and beautiful, can be applied in other situations as well. Thanks $\endgroup$ Commented Mar 19, 2020 at 15:33

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.