Circle Packing in Cone Texture

Question

This is a cone with texture generated by PatternFilling in plane.

The problem this texture mapping is not conformal.

So my question is how to generated a pattern image which could suit a fixed height cone.

So my goal is Macaroon Tower

First goal is texture level, we just generate a suitable texture with mathematica or any other softwares or methods.

Second goal is Model Level, maybe another question.

cvgmt · Accepted Answer · 2020-12-17T14:04:11.307

9

I found two ways to do this. Here only post one of them since the other one need more time to modify. I will post the other one before Christmas Eve.

All of this use the isometry between cone and sector.

The key is : Both of TextureCoordinateFunction and MeshFunctions are using Polar Coordinate {ρ*Cos[θ], ρ*Sin[θ]}.( It took me three days to find out this)

g = Graphics[{Red, Disk[{0, 0}, 1]}, PlotRangePadding -> 0];
R = 8;
α = 0.05 π;(* 0< 2α < π *)
f[ρ, θ] := {
   ρSin[α] Cos[θ/Sin[α]],
   ρSin[α] Sin[θ/Sin[α]],
   ρCos[α]
};
 ParametricPlot3D[
 f[ρ, θ], {ρ, -R, 0}, {θ, 0, 
  2 πSin[α]}, Mesh -> None, PlotStyle -> Texture[g], 
 TextureCoordinateScaling -> False,
 TextureCoordinateFunction -> 
  Function[{x, y, 
    z, ρ, θ}, {ρCos[θ], ρSin[θ]}],
 PlotPoints -> 50, ViewPoint -> {1, 1, 1}, 
 AspectRatio -> Automatic, ViewProjection -> "Orthographic", 
 Boxed -> False, Axes -> False]

edited Dec 17 '20 at 14:04

answered Dec 17 '20 at 13:55

cvgmt

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+1. Very cool. (I would have shown, or mentioned, the backside. Unless there's a known solution to circle packing on a cone, I expect this will be hard.)( – Michael E2 Dec 17 '20 at 14:04
@MichaelE2 Thanks! The other way is use MeshFunctions -> Function[{x, y, z, ρ, θ},Norm[ρ {Cos[θ], Sin[θ]} - ρ0 {Cos[θ0], Sin[θ0]}] - .4] , I will explain the methods later. – cvgmt Dec 17 '20 at 14:11
Wow, exciting. Let me play with it. If you have time, you can also play the question of Model Level. In that situation, the circles should be plane disks. Something are slight different in packing. My question was just originated by Christmas Tree. I was using RegionPoints to generating Tree, meanwhile I recall this old idea. – HyperGroups Dec 18 '20 at 07:56

cvgmt · Answer 2 · 2020-12-25T14:55:54.923

Sorry for too late to modify the another code. The idea is draw some circle or semicircle in the sector using polar coordinate.

R = 10.5;
α = 0.05 π;(* 0< 2α < π *)
r = 0.5;
ϕ = 2 π*Sin[α]; 
draw2d[k_, θ0_] := 
 With[{ρ0 = 2 k*r}, 
  ParametricPlot[{ρ*Cos[θ], ρ*
     Sin[θ]}, {ρ, 0, R}, {θ, 0, 
    2 π*Sin[α]}, 
   MeshFunctions -> 
    Function[{x, y, ρ, θ}, 
     Norm[ρ {Cos[θ], 
          Sin[θ]} - ρ0 {Cos[θ0], Sin[θ0]}] -
       r], Mesh -> {{0}}, MeshShading -> {Red, None}, 
   PlotPoints -> 80]];
Show[draw2d[0, 0], 
 Table[draw2d[k, θ0], {k, 1, Floor[R]}, {θ0, 
   Subdivide[0, ϕ, Round[ϕ/(2 ArcSin[1/(2 k)])]]}]]

And then use the isometry between sector and cone by this maps.

f[ρ_, θ_] = {ρ*Sin[α] Cos[θ/Sin[α]], ρ*Sin[α] Sin[θ/Sin[α]], ρ*Cos[α]};

After that we can lift the 2D to 3D.

R = 11.5;
α = 0.05 π;(*0<2α<π*)r = 0.5;
ϕ = 2 π*Sin[α];
f[ρ_, θ_] = {ρ*
    Sin[α] Cos[θ/Sin[α]], ρ*
    Sin[α] Sin[θ/Sin[α]], ρ*Cos[α]};
colors = {Pink, CMYKColor[4/100, 7/100, 19/100, 0]};
draw3d[k_, θ0_] := 
  With[{ρ0 = 2 k*r}, 
   ParametricPlot3D[
    f[ρ, θ], {ρ, 0, R}, {θ, 0, 
     2 π*Sin[α]}, 
    MeshFunctions -> 
     Function[{x, y, z, ρ, θ}, 
      Norm[ρ {Cos[θ], 
           Sin[θ]} - ρ0 {Cos[θ0], 
           Sin[θ0]}] - r], Mesh -> {{0}}, 
    MeshShading -> {colors[[Mod[k, 2, 1]]], None}, 
    PlotPoints -> 80]];
Show[draw3d[0, 0], 
 Table[draw3d[k, θ0], {k, 1, Floor[R]}, {θ0, 
   Subdivide[0, ϕ, Round[ϕ/(2 ArcSin[1/(2 k)])]]}], 
 Boxed -> False, Axes -> False, ViewPoint -> {-2.39, 1.72, -1.64}, 
 ViewVertical -> {-0.31, 0.18, -0.93}]

Circle Packing in Cone Texture

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