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I try to plot the following inequalities f[x,y,z]<=1 and g[x,y,z]<=1 in a single plot such that both the regions are visible and also the intersection is highlighted. I tried the RegionPlot3D but it only shows the region which is common to them.

f[x_, y_, z_] = x + y + z;
g[x_, y_, z_] = x^2 + y^2 + z^2;

RegionPlot3D[f[x, y, z] <= 1, {x, 0, 1}, {y, 0, 1}, {z, 0, 1}]
RegionPlot3D[g[x, y, z] <= 1, {x, 0, 1}, {y, 0, 1}, {z, 0, 1}]
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  • $\begingroup$ One of those regions is completely within the other, tho; is something like RegionPlot3D[{x + y + z <= 1, x^2 + y^2 + z^2 <= 1}, {x, 0, 1}, {y, 0, 1}, {z, 0, 1}, PlotStyle -> Opacity[1/3]] what you had in mind? $\endgroup$
    – J. M.'s missing motivation ♦
    Commented Jul 27, 2022 at 17:21

1 Answer 1

Reset to default
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Clear["Global`*"]

f[x_, y_, z_] = x + y + z;
g[x_, y_, z_] = x^2 + y^2 + z^2;

Since the region f[x, y, z] <= 1 is completely contained in the region g[x, y, z] <= 1, the intersection is just the smaller region. You cannot distinguish the smaller region from the intersection.

RegionPlot3D[{f[x, y, z] <= 1, g[x, y, z] <= 1, 
  f[x, y, z] <= 1 && g[x, y, z] <= 1}, {x, 0, 1}, {y, 0, 1}, {z, 0, 1},
 PlotStyle -> {Opacity[0.5], Opacity[0.5], Automatic},
 PlotLegends -> "Expressions"]

enter image description here

However, if the neither region is completely contained by the other

RegionPlot3D[{f[x, y, z] <= 1, g[x, y, z] <= 1/2, 
  f[x, y, z] <= 1 && g[x, y, z] <= 1/2}, {x, 0, 1}, {y, 0, 1}, {z, 0, 1},
 PlotStyle -> {Opacity[0.5], Opacity[0.5], Automatic},
 PlotLegends -> "Expressions"]

enter image description here

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  • $\begingroup$ Thanks Bob, it worked fine when I copied it first time, but now it is showing "{x+y+z<=1,x^2+y^2+z^2<=1,x+y+z<=1&&x^2+y^2+z^2<=1} must be a Boolean \ function"......don't really understand what's wrong here! $\endgroup$
    – seeker
    Commented Jul 27, 2022 at 19:51
  • $\begingroup$ Quit Mathematica and restart it. Copy, paste, and evaluate the code above. $\endgroup$
    – Bob Hanlon
    Commented Jul 27, 2022 at 19:55
  • $\begingroup$ Still showing the same message :-( $\endgroup$
    – seeker
    Commented Jul 27, 2022 at 20:09
  • $\begingroup$ What Mathematica version are you using? $\endgroup$
    – Bob Hanlon
    Commented Jul 27, 2022 at 20:10
  • $\begingroup$ 11.3, on Win 10 $\endgroup$
    – seeker
    Commented Jul 27, 2022 at 20:16

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