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I am studying a polynomial

F[x_, y_, z_] := 2*(x^2 - 2 x*y + y^2 - y*z)^2 - y^4 - z^4.

I tried plotting this: a = ContourPlot3D[F[x, y, z] == 0, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}]

enter image description here

But I figured out that the line $ \lambda (1, 1, 1) $ also lies on my surface, because $ F[\lambda,\lambda,\lambda] = 0 $. However, if I plot that together, it doesn't look like that.

Show[Graphics3D[{Red, Thick, Line[{{-2, -2, -2}, {2, 2, 2}}]}], a]

enter image description here

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  • $\begingroup$ As I understand it, ContourPlot3D does not show sets of dimension less than two e.g. ContourPlot3D[(x - y)^2 + (y - z)^2 == 0, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}]. $\endgroup$
    – user64494
    May 3 at 16:02
  • $\begingroup$ Thank you! So do you know maybe a way to show also the sets of dimension less than two? $\endgroup$
    – skipi
    May 3 at 16:53
  • $\begingroup$ No, I don't know such a way, expect suggested by you. $\endgroup$
    – user64494
    May 3 at 17:45
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    $\begingroup$ Try RegionPlot3D[-.02 <= F[x, y, z] <= .02, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}, PlotPoints -> 100, MaxRecursion -> 4 ] which gives roughly what you expect. $\endgroup$ May 3 at 20:03
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Workaround using RegionPlot3D

F[x_, y_, z_] := 2*(x^2 - 2 x*y + y^2 - y*z)^2 - y^4 - z^4
grad = Grad[F[x, y, z], {x, y, z}]

RegionPlot3D[-.02 Sqrt[grad . grad ] <=F[x, y, z] <= .02 Sqrt[grad . grad ]
, {x, -2, 2}, {y, -2,2}, {z, -2, 2} , PlotPoints -> 100, MaxRecursion -> 4]

enter image description here

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