RegionPlot3D KnotData

Question

Is it possible to find or construct a RegionPlot out of KnotData? My motivation is to Texture the knots as per here.

After much searching and playing around, I found this, which frustratingly doesn't give the equations for the inTinftube and onTinftube functions.

Desired output is something like this (but in 3D):

Answer 1

Edit

n = 15;
vor = VoronoiMesh[
   RandomPoint[Rectangle[{0, 0}, {2 π, 2 π}], 
    n], {{0, 2 π}, {0, 2 π}}];
polys = MeshPrimitives[vor, 2];
g = Show[Table[{Red, 
      Disk[x /. Last[#], Abs@First[#]] &@
       NMinimize[SignedRegionDistance[poly][x], 
        x ∈ poly]}, {poly, polys}] // Graphics];
curve3 = KnotData["Trefoil", "SpaceCurve"];
basis = Last[FrenetSerretSystem[curve3[t], t]];
{tangent, normal, binormal} = basis;
ParametricPlot3D[
 curve3[t] + .6 (Cos[u]*normal + Sin[u]*binormal), {u, 0, 
  2 π}, {t, 0, 2 π}, PlotPoints -> 80, Mesh -> None, 
 Boxed -> False, Axes -> False, PlotStyle -> Texture[g], 
 TextureCoordinateScaling -> True, 
 TextureCoordinateFunction -> Function[{x, y, z, u, t}, {u, 9 t}], 
 ViewPoint -> {0.2, -0.3, 3.3}]

Original

A starting point.

curve3 = KnotData["Trefoil", "SpaceCurve"];
basis = Last[FrenetSerretSystem[curve3[t], t]];
{tangent, normal, binormal} = basis;
g = Graphics[{Red, Disk[{0, 0}, .5]}, PlotRangePadding -> .5];
ParametricPlot3D[
 curve3[t] + .6 (Cos[u]*normal + Sin[u]*binormal), {u, 0, 
  2 π}, {t, 0, 2 π}, PlotPoints -> 80, Mesh -> None, 
 Boxed -> False, Axes -> False, PlotStyle -> Texture[g], 
 TextureCoordinateScaling -> False, 
 TextureCoordinateFunction -> Function[{x, y, z, t, u}, {x, y}], 
 ViewPoint -> {0.2, -0.3, 3.3}]

Answer 2

You can feed the BoundaryMeshRegion[] you can obtain from KnotData[] into RegionPlot3D[]. For example:

trefBMR = KnotData["Trefoil", "BoundaryMeshRegion"];
RegionPlot3D[RegionMember[trefBMR, {x, y, z}],
             {x, -7/2, 7/2}, {y, -7/2, 7/2}, {z, -3/2, 3/2},
             BoxRatios -> Automatic, Lighting -> "Neutral",
             Mesh -> None, PlotPoints -> 35, 
             PlotStyle -> Directive[Texture[ExampleData[{"ColorTexture",
                                                         "WhiteMarble"}]]]]

Answer 3

Multicolumn[Graphics3D[{SurfaceAppearance["TextureShading", Texture[disks]], 
     Tube[BSplineCurve[KnotData[#, "SpaceCurve"] /@ Subdivide[0, 2 Pi, 100], 
       SplineClosed -> True], .4]}, 
    Boxed -> False, ImageSize -> Medium, ViewPoint -> {0, 0, 5}] & /@ 
 {{"PretzelKnot", {5, 3, 2}}, "FigureEight",
  {"TorusKnot", {3, 5}}, {"TorusKnot", {4, 9}}}, 2] 

Update: In versions 12.1+, we can use the directive SurfaceAppearance["TextureShading", Texture[img]] to texturize any surface with img:

reg = TriangulateMesh[BoundaryDiscretizeRegion[Rectangle[]], MaxCellMeasure -> .02];
disks = Graphics[{Red, MeshPrimitives[reg, 2] /. Polygon -> (Apply[Disk] @* Insphere)}];
Graphics3D[{SurfaceAppearance["TextureShading", Texture[disks]], 
   KnotData["Trefoil", "ImageData"]}, 
  Boxed -> False, ImageSize -> Large] 

We can construct a Tube with the desired radius using KnotData["Trefoil", "SpaceCurve"]:

Graphics3D[{SurfaceAppearance["TextureShading", Texture[disks]], 
  Tube[BSplineCurve[KnotData["Trefoil", "SpaceCurve"] /@ Subdivide[0, 2 Pi, 100], 
    SplineClosed -> True], .5]}, 
 Boxed -> False, ImageSize -> Large] 

Alternatively, we can use SurfaceAppearance["TextureShading", Texture[disks]] as the setting for PlotStyle in ParametricPlot3D in cvgmt's answer:

ParametricPlot3D[curve3[t] + .6 (Cos[u] normal + Sin[u] binormal), 
 {u, 0, 2 π}, {t, 0, 2 π}, PlotPoints -> 80, Mesh -> None, 
 Boxed -> False, Axes -> False, 
 PlotStyle -> SurfaceAppearance["TextureShading", Texture[disks]], 
 ViewPoint -> {0.2, -0.3, 3.3}, Lighting->"Neutral"]

Original answer:

We can use the new-in-12.1 directive HalfToneShading:

Graphics3D[{HalftoneShading[#, Red], KnotData["Trefoil", "ImageData"]}, 
 Lighting -> "Neutral", ImageSize -> 250, Boxed -> False, 
    ViewPoint -> {1.5, -1.5, 4.}] & /@ {.3, .5, .7} // Row 

Needless to say, this approach is not match for cvgmt's approach in terms of flexibility and beauty of the pictures produced.

To get some flexibility in controlling the density of shapes, we can use the options of SurfaceAppearance to define a directive with options:

Options[surfaceAppearance] = {"StepCount" -> 1, "Tiling" -> {5, 5}, 
   "FeatureColor" -> Red, "UseScreenSpace" -> 0, "IsTwoTone" -> 1, 
   "LuminanceModifier" -> 0.0, "Shape" -> "Disk"};
surfaceAppearance[opts : OptionsPattern[surfaceAppearance]] := 
 SurfaceAppearance["RampShading", 
  Sequence @@ FilterRules[{opts, Options[surfaceAppearance]}, Except["Shape"]], 
  "Arguments" -> {"HalftoneShading", 0.5, Red, OptionValue["Shape"]}, 
  EdgeForm[], Texture["HalftoneShading" <> OptionValue["Shape"]]]

Examples:

Graphics3D[{surfaceAppearance[], KnotData["Trefoil", "ImageData"]},
  Lighting -> "Accent", Boxed -> False, ViewPoint -> {1.5, -1.5, 4.}]

Use surfaceAppearance["Tiling" -> {15, 15}] to get:

Use surfaceAppearance["UseScreenSpace" -> 1, "StepCount" -> 2, "Tiling" -> {7, 7}] to get:

Use surfaceAppearance["Tiling" -> {15, 15}, "Shape"->"Triangle"] to get:

Use surfaceAppearance["StepCount" -> 3,"Tiling" -> {10,10},"Shape" -> "Hexagon"] to get:

Answer 4

SliceContourPlot3D[Sin[5 x] Sin[6 y] Sin[4 z], 
 KnotData["Trefoil", "Region"],
 {x, -Pi, Pi}, {y, -Pi, Pi}, {z, -Pi, Pi}, 
 Contours -> {-1/6, 1/6}, ContourStyle -> None, 
 ContourShading -> {White, Red}, 
 Boxed -> False, ImageSize -> Large, Axes -> False, PlotPoints -> 90]