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Writing:

r[k_] = {k, 2 Sqrt[6] Cos[u], 2 Sqrt[6] Sin[u]};
ParametricPlot3D[r /@ {-1, 1}, {u, 0, 2 Pi}]

I get:

enter image description here

and this is fine. On the other hand, writing:

A = ImplicitRegion[3 x^2 + y^2 + z^2 == 27 && x^2 + y^2 + z^2 == 25, {x, y, z}];
RegionPlot3D[A]

I get:

enter image description here

and I do not understand why. What am I doing wrong?

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  • 3
    $\begingroup$ Unless you use some special regions the region plot is still based on sampling and it is tough to get informative results for 1D region in 3D. So you have to improvise like in at least closely related: mathematica.stackexchange.com/q/5968/5478 The accepted answer works almost out of hand. $\endgroup$ – Kuba Mar 10 '17 at 10:00
  • $\begingroup$ Thank you! However, I can not understand how I could do then to plot the colors in the same way as I did here: mathematica.stackexchange.com/questions/136210/… $\endgroup$ – TeM Mar 10 '17 at 22:31
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I think it is simply because you said == 27 there. The region is simply too small to show. very thin region! (line thin)

You could try

a=ImplicitRegion[3 x^2+y^2+z^2<=27 ,{x,y,z}];
b=ImplicitRegion[x^2+y^2+z^2<=  25 ,{x,y,z}];
Show[RegionPlot3D[a,PlotStyle->Red],RegionPlot3D[b,PlotStyle->Blue]]

Mathematica graphics

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Reparametrizing:

f[u_, v_] := Sqrt[27] {Sin[u] Cos[v]/Sqrt[3], Sin[u] Sin[v], Cos[u]}
g[u_, v_] := 5 {Sin[u] Cos[v], Sin[u] Sin[v], Cos[u]}

From inspection the cartesian definitions intersect at circles in y-z plane at$x=\pm 1$. This can be done:

(* the 2 surfaces *)
p = ParametricPlot3D[{f[u, v], g[u, v]}, {u, 0, Pi}, {v, 0, 2 Pi}, 
Mesh -> None, PlotPoints -> 50]
(* the cartesian definition *)
f[x_, y_, z_] := 3 x^2 + y^2 + z^2 - 27
g[x_, y_, z_] := x^2 + y^2 + z^2 - 25
(* visualization *)
Show[p, ParametricPlot3D[{x, Cos[u] Sqrt[25 - x^2], 
    Sin[u] Sqrt[25 - x^2]} /. 
   Quiet@Solve[f[x, y, z] == g[x, y, z], {x, y, z}], {u, 0, 2 Pi}, 
  PlotStyle -> Red]]

enter image description here

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