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I'm having troubles plotting an implicit region R (the full expression will be given at the end of this message for sake of readability). This implicit region R is defined by both inequalities and equalities, for which I can separately use RegionPlot3D and ContourPlot3D without any problems. However, when I want to have a plot of the sole region where these two sets intersect, I have to use another command, and in particular I tried:

  • Region[R]: returns an empty output.
  • DiscretizeRegion[R]: returns the message "DiscretizeRegion was unable to discretize the region ImplicitRegion".
  • Needs["NDSolveFEM "], ToElementMesh[R]: returns the message "BoundaryDiscretizeRegion: There is not a boundary representation that uniquely defines a region with region dimension 2 embedded in dimension 3".

I don't know what else to try. From the previously cited combination of RegionPlot3D and ContourPlot3D I see that the region exists and it should also be quite smooth to plot, so I really don't know what the problem could be. Any help would be highly appreciated!

The region R is the following:

ImplicitRegion[-((
    S^2 (-1 + K + S) (S + (-1 + K) T))/(-T + (K + S) (S + K T))) > 0 &&
   2 S > 0 && (1/((T - (K + S) (S + K T))^2))
    2 S^2 (S^2 (1 + S (K + S)^2) + 
       S (-2 - S + (K + S) (-S + 2 (K + K S (K + S)))) T + (-1 + 
          K (K + S)) (-2 + K (2 + S (K + S))) T^2) - (
    1/((T - (K + S) (S + K T))^2))
    S^2 (-S ((-1 + K) K + 2 (-1 + K) S + S^2) - (K + K^3 + 
          K^2 (-2 + S) + 3 S - S^3 - K S (4 + S)) T + (2 - S + 
          K (-3 - S + (K + S)^2)) T^2) > 
   0 && (1/((T - (K + S) (S + K T))^4))
    2 S^4 (S^2 (1 + S (K + S)^2) + 
       S (-2 - S + (K + S) (-S + 2 (K + K S (K + S)))) T + (-1 + 
          K (K + S)) (-2 + K (2 + S (K + S))) T^2) (-S ((-1 + K) K + 
          2 (-1 + K) S + S^2) - (K + K^3 + K^2 (-2 + S) + 3 S - S^3 - 
          K S (4 + S)) T + (2 - S + K (-3 - S + (K + S)^2)) T^2) - (
    1/((T - (K + S) (S + K T))^4))
    S^4 (-S ((-1 + K) K + 2 (-1 + K) S + S^2) - (K + K^3 + 
          K^2 (-2 + S) + 3 S - S^3 - K S (4 + S)) T + (2 - S + 
          K (-3 - S + (K + S)^2)) T^2)^2 + (
    4 S^4 (-1 + K + S) (S + (-1 + K) T))/(-T + (K + S) (S + K T)) == 
   0 && ((K < -1 && ((0 < S < (1 - K^2)/K && -(S/(-1 + K)) < 
           T < (-K S - S^2)/(-1 + K^2 + K S)) || ((1 - K^2)/K <= S < 
           1 - K && T > -(S/(-1 + K))) || (S > 1 - K && 
          0 < T < -(S/(-1 + K))))) || (-1 <= K < 
       1 && ((0 < S < 1 - K && T > -(S/(-1 + K))) || (S > 1 - K && 
          0 < T < -(S/(-1 + K)))))) && -1.5 <= K <= 1 && 
  0.001 <= S <= 2 && 0.001 <= T <= 3, {K, S, T}]
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  • $\begingroup$ try RegionPlot3D i hope you code neat and nice so i could help further ! $\endgroup$ – Alrubaie Apr 3 at 17:08
  • $\begingroup$ Unfortunately RegionPlot3D doesn't work because, as I wrote, there are both inequalities and equalities in the definition of the region, and RegionPlot3D only accepts inequalities. $\endgroup$ – annuk89 Apr 3 at 17:11
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I was able to get the region to discretize over larger bounds. So one workaround is to clip this discretization:

DiscretizeRegion[
  DiscretizeRegion[R, {{-1.5, 1}, {0, 2}, {0, 3}}, MaxCellMeasure -> .00001], 
  {{-1.5, 1}, {0.001, 2}, {0.001, 3}}
]


With more work, we can invoke ContourPlot3D:

With[
  {
    equality = Numerator[Together[FirstCase[R[[1]], _Equal][[1]]]] == 0, 
    cons = R[[1]] /. {_Equal -> True, K -> #1, S -> #2, T -> #3}
  },
  ContourPlot3D[{equality}, {K, -1.5, 1}, {S, 0.001, 2}, {T, 0.001, 3}, 
    PlotPoints -> 60, RegionFunction -> Function[cons]]
]

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  • $\begingroup$ This works perfectly. Thank you! :) $\endgroup$ – annuk89 Apr 3 at 17:51
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You could use DiscretizeRegion:

DiscretizeRegion[R, {{-1.5,1},{0,2},{0,3}}]

enter image description here

where R is your ImplicitRegion, and the bounds coming from your last 3 inequalities.

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  • $\begingroup$ Thank you for your answer. This seems to work, however the two variables S and T should be greater than 0, and as soon as I take for instance {0.001,2},{0.001,3} in the DiscretizeRegion, I get again an error message - it only works if I put 0 instead (but then somehow these horizontal/vertical areas are not convincing). Any clue what could the issue be? $\endgroup$ – annuk89 Apr 3 at 17:34

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