Thick polygons in Graphics3D [duplicate]

Question

This question already has an answer here:

• How to make hollow polyhedra? 3 answers

I have a polygon that I want to thicken with thickness h. How do I do this?

FaceForm is evil and says:

The directive Thickness was encountered in a context where it is not allowed.

Answer 1

Similar to ssch's method, but I though I'd throw in mine, it' just finds the normal of the polygon (The average normal if the polygon isn't planar). It then moves the polygon along this direction to create a top and bottom polygon, and then fill in the sides between these two:

normal[a_,b_,c_]:=Normalize@Cross[a-b,c-b]
normal[a___]:=Mean[normal@@@Partition[{a},3,1,1]]

sides[bottom_,top_] :=     
   Polygon[Reverse@Join[#1,Reverse@#2]]&@@@({bottom,top}//Transpose//Partition[#,2,1,1]&)

thicken[val_,t_:0.1]:=val/.Polygon[bottom_,___]:>
 With[{top=(# +t normal@@bottom)&/@bottom},
    {Polygon[Reverse@bottom],
     sides[bottom,top],
     Polygon[top]
    } 
 ]   

Applying multiple times can lead to fun results:

initial = 
Graphics3D[{Arrow@{{0, 0, 0}, {1, 0, 0}, {0, 1, 0}}, 
Polygon[{{0, 0, 0}, {1, 0, 0}, {0, 1, 0}}]}];
NestList[thicken[#, 0.3] &, initial, 2] // GraphicsRow

To use this with graphics generated by for instance Plot3D, you simply have to use Normal to expand out GraphicsComplex's. I also added a slight pattern above (The second part of Polygon[bottom_,___]) which makes the code ignore anything in polygon after the points. This is nessesary because some graphics include normals which are not accounted for above:

(Normal@Plot3D[Cos[x] Cos[y], {x, 0, \[Pi]}, {y, 0, \[Pi]}, 
 Mesh -> None, PlotPoints -> 2]) // thicken[#, 0.3] &

Answer 2

Layering:

By creating a bunch of the same polygon at slightly different heights on top of each other the effect can be simulated:

(* Same as Table but high is always included*)
Attributes[tableEnsureUpper]=HoldAll;
tableEnsureUpper[expr_, {sym_, low_, high_, step_: 1}] := 
 If[Mod[high - low, step, step] == step,
  Table[expr, {sym, low, high, step}],
  Append[Table[expr, {sym, low, high, step}],Block[{sym=high}, expr]]
  ]
(* Expects 2D Polygon in *)
raisePolygonLayered[p_Polygon, h_: 1] := 
 tableEnsureUpper[
  p /. {a_?NumericQ, b_?NumericQ} :> {a, b, z}, {z, 0, h, 0.2}]

Problems is that the slices are visible from the side unless very many are used.

Edge Polygon

The method is based on taking each successive (overlapping) pair of points and creating the corresponding rectangle with height h:

raisePolygonEdge[p_Polygon, h_: 1] := Module[{
   data = p[[1]],
   pts
   },
  If[Depth[data] == 3, data = {data}];
  pts = Flatten[Partition[Riffle[Most@#, Rest@#], 2] & /@ data, 1];
  Polygon[ Join[{ 
       {#[[1, 1]], #[[1, 2]], 0}, {#[[1, 1]], #[[1, 2]], h},
       {#[[2, 1]], #[[2, 2]], h}, {#[[2, 1]], #[[2, 2]], 0}
       } & /@ pts, data /. {a_?NumericQ, b_?NumericQ} :> {a, b, h}]]
  ]

Answer 3

I've already answered something like this here, but here's a slightly simplified version you might find useful:

(* barycenter of a polygon *)
averagepoints[points_?MatrixQ] :=
        Mean[If[TrueQ[First[points] == Last[points]], Most, Identity][points]]

(* Newell's algorithm for face normals *)
newellNormals[pts_?MatrixQ] := Module[{tp = Transpose[pts]}, Normalize[MapThread[Dot,
              {RotateLeft[ListConvolve[{{-1, 1}}, tp, {-1, -1}]], 
               RotateRight[ListConvolve[{{1, 1}}, tp, {-1, -1}]]}]]]

thickenaux[points_, thick_] := Module[{center = averagepoints[points],
  nrm = newellNormals[points], outerpoints, radialpoints}, 
  outerpoints = Map[TranslationTransform[thick nrm], points];
  radialpoints = MapThread[Join[#1, Reverse[#2]] &,
                           Map[Partition[#, 2, 1, 1] &, {points, outerpoints}]];
  Flatten[{Polygon[Reverse[points]], Polygon /@ radialpoints, Polygon[outerpoints]}]]

ThickenPolygons[shape_, thick_: 0.04] :=
       shape /. Polygon[p_?MatrixQ] :> thickenaux[p, thick]

Examples:

NestList[ThickenPolygons,
         Graphics3D[Polygon[{{0, 0, 0}, {1, 0, 0}, {0, 1, 0}}]], 2] // GraphicsRow

PolyhedronData["Tetrahedron"] // Normal // ThickenPolygons