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In the following graphic of a solid torus, how can I show in the plane z = 0 just the outer "equatorial circle", and not the smaller circle inside, without parameterizing that equatorial circle?

With[{R = 2, r = 0.5}, 
 Show[{RegionPlot3D[(R - Sqrt[x^2 + y^2])^2 + z^2 <= r^2, {x, -3, 
     3}, {y, -3, 3}, {z, -1, 1}, MaxRecursion -> 5, PlotPoints -> 60, 
    PlotStyle -> Opacity[0.2], Mesh -> None, AxesOrigin -> {0, 0, 0}, 
    Ticks -> None, Boxed -> False, BoxRatios -> {1, 1, 0.5}],
   SliceContourPlot3D[(R - Sqrt[x^2 + y^2]) + 
     r, {"ZStackedPlanes", {0}}, {x, -3, 3}, {y, -3, 3}, {z, -1, 1},
    Contours -> 1, ContourShading -> None,
    ContourStyle -> Directive[Thick, Red], AxesOrigin -> {0, 0, 0}, 
    Ticks -> None, Boxed -> False]}, ImageSize -> Scaled[0.7]]
 ]

Solid torus with equator and spurious circle

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  • $\begingroup$ Don't use a contour plot, use this: ResourceFunction["Circle3D"] in a Graphics3D $\endgroup$
    – flinty
    Commented Mar 17 at 15:22
  • $\begingroup$ Add BoundaryStyle -> None, RegionFunction -> Function[{x, y, z, f}, Sqrt[x^2 + y^2] > r] into SliceContourPlot3D. $\endgroup$
    – cvgmt
    Commented Mar 17 at 15:25
  • $\begingroup$ Simple solution: Graphics3D[{{Opacity[0.5], Torus[{0, 0, 0}, {5, 10}]}, Black, Torus[{0, 0, 0}, {10, 10.2}]}]`` $\endgroup$
    – Daniel Huber
    Commented Mar 17 at 17:28
  • $\begingroup$ Solution using RegionFunction is closest in spirit to my original. The ResourceFunction["Circle3D"] seems awkward in that default has circle in plane perpendicular to x-axis. The solution using Torus sort of defeats my purpose in actually using equation of the torus, and its equatorial circle becomes a tube. $\endgroup$
    – murray
    Commented Mar 17 at 19:21
  • $\begingroup$ How about plot /. {dir_, l1_Line, l2_Line} :> {dir, l1} to remove the offending line? $\endgroup$
    – Michael E2
    Commented Mar 17 at 20:00

1 Answer 1

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You can use RegionFunction and BoundaryStyle:

With[{R = 2, r = 0.5},
 Show[{
   RegionPlot3D[
    (R - Sqrt[x^2 + y^2])^2 + z^2 <= r^2
    , {x, -3, 3}, {y, -3, 3}, {z, -1, 1}
    , MaxRecursion -> 5, PlotPoints -> 60, PlotStyle -> Opacity[0.2], 
    Mesh -> None, AxesOrigin -> {0, 0, 0}, Ticks -> None, 
    Boxed -> False, BoxRatios -> {1, 1, 0.5}],
   SliceContourPlot3D[
    (R - Sqrt[x^2 + y^2]) + r
    , {"ZStackedPlanes", {0}}
    , {x, -3, 3}, {y, -3, 3}, {z, -1, 1}
    , Contours -> 1, ContourShading -> None, 
    ContourStyle -> Directive[Thick, Red], AxesOrigin -> {0, 0, 0}, 
    Ticks -> None, Boxed -> False, 
    RegionFunction -> Function[{x, y, z}, x^2 + y^2 > R^2], 
    BoundaryStyle -> None
    ]}
  , ImageSize -> Scaled[0.7]]]

enter image description here

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