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plotting - Smooth ParametricPlot3D with RegionFunction?


What's the preferred way to paint spots on a curved surface without getting raggedy edges? Using ParametricPlot3D with RegionFunction gets the shape I want but increasing PlotPoints slows things down considerably. Counterintuitively Reducing PlotPoints to 50 gives better looking results than 100:



With[{a = 0.8, b = 0.4, r = 5^0.5, l = 10, points = 50}, 
 ParametricPlot3D[{{x, -Sqrt[r^2 - x^2], z}, {x, Sqrt[r^2 - x^2], 
    z}}, {x, -r, r}, {z, -l/2, l/2}, 
  RegionFunction -> 
   Function[{x, y, z}, (x/a)^2 + (z/b)^2 < 1 && y > 0], Mesh -> None, 
  PerformanceGoal -> "Quality", PlotPoints -> points]]

First image has points =50, second image has points = 100.


enter image description here enter image description here



Answer





What's the preferred way to paint spots on a curved surface without getting raggedy edges?



I use MeshFunctions, MeshShading, and Mesh:


Show[
 (* Head *)
 ParametricPlot3D[
  (1 + 0.05 Cos[u]) {Sin[u] Cos[v], Sin[u] Sin[v], 0} + {0, 0, 1.05 Cos[u]},
  {u, 0, π}, {v, 0, 2 π},
  (* mouth *)

  MeshFunctions -> {Function[{x, y, z, u, v}, x + 0.5 (2 y^2 - 2 (z + 0.5))^2]},
  MeshShading -> {Black, Lighter@Red, Yellow},
  Mesh -> {{-0.85, -0.82}},
  (*****)
  AxesLabel -> {"x", "y", "z"}, PlotPoints -> 100],
 (* Eyes *)
 ParametricPlot3D[
  Sin[1.9 (u - 0.45)]^2 (0.36 + 0.8 (Sqrt[0.5 + 2 Abs[v]])) Cos[1.5 v] *
     {-Sin[u] Cos[v], -Sin[u] Sin[v], Cos[u]},
  {u, π/6, π/2}, {v, -π/6, π/6},

  (* pupil & iris *)
  MeshFunctions -> {Function[{x, y, z, u, v}, 
     Sin[1.9 (u - 0.49)]^2 (0.36 + 0.8 (Sqrt[0.5 + 2 Abs[v]])) Cos[1.5 v]]},
  MeshShading -> {White, Lighter@Blue, Black},
  Mesh -> {{1.06, 1.077}},
  (*****)
  AxesLabel -> {"x", "y", "z"}, PlotPoints -> 100],
 (* Tongue *)
 ParametricPlot3D[
  {-u, 0, -u^2/4 - 0.15} + 0.15 Sqrt[Sqrt[(1.2 - u)]] *

     (2 Cos[v] {0, 1, 0} + Sin[v] {-u/2, 0, 1 - 0.5 Sin[v]}/Sqrt[u^2 + 1]),
  {u, 0.5, 1.2}, {v, 0, 2 π}, PlotStyle -> Red, Mesh -> None],
 PlotRange -> All
 ]

Mathematica graphics


It has the advantage of having borders that meet each other exactly. (Pasting a surface patch on another surface usually requires a small gap between them, or rounding error in the GPU causes the image to shimmer.)


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