Find the area of a contour surface plotted by ContourPlot3D

Question

I am looking for a way to find the surface area of a contour from ContourPlot3D. Here is an example where I might like to find the area of the contour surface to G that equals -5:

G[x_, y_, z_] := 
  -2 Log[(2 + 2 Sqrt[((y)^2 + (z)^2)^2 + (-1 - Abs[x])^2] + 2  Abs[x])/
         (-2 + 2 Sqrt[((y)^2 + (z)^2)^2 + (1 - Abs[x])^2] + 2 Abs[x])] - 
   2 Log[(1. + 2 Sqrt[((x)^2 + (y)^2)^2 + (-0.5 - Abs[-0.5 + z])^2] + 
            2 Abs[-0.5 + z])/
         (-1. + 2 Sqrt[((x)^2 + (y)^2)^2 + (0.5 - Abs[-0.5` + z])^2] + 
            2 Abs[-0.5 + z])]

ContourPlot3D[G[x, y, z], {x, -2, 2}, {y, -2, 2}, {z, -2, 2}, 
  Contours -> {-5}]

I would prefer a general method that would work even with more complicated functions than this. I don't know if there is a way to derive the area from the contour plot itself, or some other way accomplish this. Any ideas?

Answer 1

This problem can be solved out of the box in V10 using the ImplicitRegion function. Lets use your function G and define the bounded 3D region. We can use RegionPlot3D to visualize the object.

region = ImplicitRegion[G[x, y, z] <= -5, {x, y, z}];
RegionPlot3D[region, PlotRange -> 1.5, PlotPoints -> 60]

Now we will use DiscretizeRegion to form a MeshRegion. The RegionBoundary will give the 3D surface mesh from which we can easily extract the surface area using the Areaproperty.

dis = DiscretizeRegion[region,{{-1.5, 1.5},{-1.5,1.5},{-1.5, 1.5}},
AccuracyGoal -> 9,MaxCellMeasure -> {"Area" -> 0.01}];
HighlightMesh[dis, Style[1, Thin, Red]]

Area@RegionBoundary[dis]

18.7526

However a considerably slower Method -> "ContourPlot3D" will generate a more accurate mesh resulting in a more precise area calculation.

disFine = DiscretizeRegion[region,{{-1.5, 1.5},{-1.5, 1.5},{-1.5,1.5}},
Method -> "ContourPlot3D",AccuracyGoal -> 9,MaxCellMeasure -> {"Area" -> 0.01}]

Area@RegionBoundary[disFine]

18.7894

Answer 2

You can also use DiscretizeGraphics with some clever options in your ContourPlot3D. Just add the Option Lighting -> "Neutral":

gr = ContourPlot3D[G[x, y, z], {x, -2, 2}, {y, -2, 2}, {z, -2, 2}, 
  Contours -> {-5}, Lighting -> "Neutral"] // Quiet;

Then you can compute the area with:

Area @ DiscretizeGraphics @ gr

18.8018

Here is the image from DiscretizeGraphics: