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Function Repository Resource:

OutlinePolygons

Source Notebook

Outline polygons by beam-like representation

Contributed by: Wolfram Research

ResourceFunction["OutlinePolygons"][gr]

outlines the polygons in the 3D graphics gr by beam-like representations.

ResourceFunction["OutlinePolygons"][gr, inner, outer]

outlines the polygons with ratios inner and outer.

Details and Options

The width of each beam is formed by creating a hole in each polygon. Holes are created by scaling each point in the polygon inward by ratio inner.
The height of each beam oriented with respect to the origin is given by scaling the polygon inward by ratio outer.
ResourceFunction["OutlinePolygons"] works best with planar convex polygons. It does not support arbitrary 3D graphics primatives.
ResourceFunction["OutlinePolygons"][gr] is equivalent to ResourceFunction["OutlinePolygons"][gr,0.8,0.9].

Examples

Basic Examples (9) 

An outlined tetrahedron:

In[1]:=
ResourceFunction["OutlinePolygons"][
 Graphics3D[
  Polygon /@ {{{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}, {{0, 1, 0}, {1, 0, 0}, {0, 0, 0}}, {{0, 0, 1}, {0, 0, 0}, {1, 0, 0}}, {{0, 0, 0}, {0, 0, 1}, {0, 1, 0}}}]]
Out[1]=

Specify the inner and outer ratios:

In[2]:=
ResourceFunction["OutlinePolygons"][\!\(\*
Graphics3DBox[
{EdgeForm[None], Polygon3DBox[{{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}], Polygon3DBox[{{0, 1, 0}, {1, 0, 0}, {0, 0, 0}}], Polygon3DBox[{{0, 0, 1}, {0, 0, 0}, {1, 0, 0}}], Polygon3DBox[{{0, 0, 0}, {0, 0, 1}, {0, 1, 0}}]},
ImageSize->{90., 90.},
ViewAngle->0.5011114127587019,
ViewPoint->{1.3, -2.4, 2.},
ViewVertical->{0., 0., 1.}]\), .5, .8]
Out[2]=

A spikey:

In[3]:=
Entity["Polyhedron", "RhombicHexecontahedron"][
  EntityProperty["Polyhedron", "Graphics3D"]] // ResourceFunction[
 "OutlinePolygons"]
Out[3]=

OutlinePolygons works with other primitives:

In[4]:=
ResourceFunction["OutlinePolygons"][
  Dodecahedron[{.1, .5}]] // Graphics3D
Out[4]=
In[5]:=
Graphics3D[
  Polyhedron[{{0., 0., 0.6}, {-0.3, -0.5, -0.2}, {-0.3, 0.5, -0.2}, {0.6, 0., -0.2}}, {{2, 3, 4}, {3, 2, 1}, {4, 1, 2}, {1, 4, 3}}]] // ResourceFunction["OutlinePolygons"]
Out[5]=

A Polyhedron with voids:

In[6]:=
(* Evaluate this cell to get the example input *) CloudGet["https://www.wolframcloud.com/obj/3a9a5ef5-ded3-48ff-b109-559a9ef8f42a"]
Out[6]=

Add a texture:

In[7]:=
vtc = {{0, 0}, {1, 0}, {1, 1}, {0, 1}};
coords = {{{0, 0, 0}, {0, 1, 0}, {1, 1, 0}, {1, 0, 0}}, {{0, 0, 0}, {1, 0, 0}, {1, 0, 1}, {0, 0, 1}}, {{1, 0, 0}, {1, 1, 0}, {1, 1, 1}, {1, 0, 1}}, {{1, 1, 0}, {0, 1, 0}, {0, 1, 1}, {1, 1, 1}}, {{0, 1, 0}, {0, 0, 0}, {0, 0, 1}, {0, 1, 1}}, {{0, 0, 1}, {1, 0, 1}, {1, 1, 1}, {0, 1, 1}}};
In[8]:=
(* Evaluate this cell to get the example input *) CloudGet["https://www.wolframcloud.com/obj/f15e073d-8bc3-4642-8036-fd18a30907e7"]
Out[8]=
In[9]:=
(* Evaluate this cell to get the example input *) CloudGet["https://www.wolframcloud.com/obj/c8e07d98-401c-428c-9153-75b84d1a7bab"]
Out[9]=

Use VertexColors:

In[10]:=
(* Evaluate this cell to get the example input *) CloudGet["https://www.wolframcloud.com/obj/9503143b-cb3d-42dc-b6a0-4bc74d8f67c4"]
Out[10]=
In[11]:=
ResourceFunction["OutlinePolygons"][
 Graphics3D[
  Polyhedron[{{0, 0, Sqrt[2/3] - 1/(2 Sqrt[6])}, {-(1/(2 Sqrt[3])), -(1/2), -(1/(
      2 Sqrt[6]))}, {-(1/(2 Sqrt[3])), 1/2, -(1/(2 Sqrt[6]))}, {1/
     Sqrt[3], 0, -(1/(2 Sqrt[6]))}}, {{2, 3, 4}, {3, 2, 1}, {4, 1, 2}, {1, 4, 3}}, VertexColors -> {Red, Green, Yellow, Blue}]]]
Out[11]=

Use VertexNormals:

In[12]:=
n = {1, -1, 1};
In[13]:=
Graphics3D[{Yellow, Polyhedron[{{0, 0, Sqrt[2/3] - 1/(2*Sqrt[6])}, {-1/(2*Sqrt[3]), -1/
       2, -1/(2*Sqrt[6])}, {-1/(2*Sqrt[3]), 1/2, -1/(2*Sqrt[6])}, {1/Sqrt[3], 0, -1/(2*Sqrt[6])}}, {{2, 3, 4}, {3, 2, 1}, {4, 1, 2}, {1, 4, 3}}, VertexNormals -> {-n, n, n, n}]}] // ResourceFunction[
 "OutlinePolygons"]
Out[13]=

Use Scaled coordinates:

In[14]:=
ResourceFunction["OutlinePolygons"][
 Graphics3D[
  Polyhedron[{{Scaled[{0.2, 0., 0.3}], Scaled[{0.2, 0.9, 0.3}], Scaled[{1, 0.5, 0.3}]}, {Scaled[{0.2, 0.9, 0.3}], Scaled[{0.2, 0., 0.3}], Scaled[{0.5, 0.5, 1.}]}, {Scaled[{1, 0.5, 0.3}], Scaled[{0.5, 0.5, 1.}], Scaled[{0.2, 0., 0.3}]}, {Scaled[{0.5, 0.5, 1.}], Scaled[{1, 0.5, 0.3}], Scaled[{0.2, 0.9, 0.3}]}}]]]
Out[14]=

Neat Examples (4) 

Create a periodic structure:

In[15]:=
uehp = Graphics3D[{#, Translate[#, {0, 1.732, 0}], Translate[#, {2, 0, 0}], Translate[#, {2, 1.732, 0}]} &@(Entity["Polyhedron", {"Prism", 6}]["Graphics3D"] // ResourceFunction["PerforatePolygons"] // ResourceFunction[
      "OutlinePolygons"])[[1]], Boxed -> False]
Out[15]=

Outline a Klein bottle:

In[16]:=
KleinBottle[{A_, B_, C_}, {s_, t_}] := {Piecewise[{{A Cos[s] (1 + Sin[s]) + B (1 - Cos[s]/2) Cos[s] Cos[t], 0 <= s <= Pi}, {A Cos[s] (1 + Sin[s]) + B (1 - Cos[s]/2) Cos[t + \[Pi]], Pi <= s <= 2 Pi}}], Piecewise[{{C Sin[s] + B (1 - Cos[s]/2) Sin[s] Cos[t], 0 <= s <= Pi}, {C Sin[s], Pi <= s <= 2 Pi}}], B (1 - Cos[s]/2) Sin[t]}
In[17]:=
ResourceFunction["OutlinePolygons"][
 With[{S = 3.415, \[Rho]s = 0.258`, \[Rho]t = 0.338`}, ParametricPlot3D[
   KleinBottle[{6, 6, 19}, {s, t}], {s, 0, 2 \[Pi]}, {t, 0, 2 \[Pi]}, Exclusions -> None, PlotStyle -> Opacity[0.8`], PerformanceGoal -> "Quality", Mesh -> None, RegionFunction -> (! (#4 - S)^2/\[Rho]s^2 + (#5 - Pi)^2/\[Rho]t^2 <
         1 &), Boxed -> False, Axes -> False]]]
Out[17]=

A 3D model of Beethoven:

In[18]:=
ResourceFunction["OutlinePolygons"][
 ExampleData[{"Geometry3D", "Beethoven"}]]
Out[18]=

Compute the circumsphere of a platonic solid, solving for edge length l, then the insphere, and let be the circumsphere of the next:

In[19]:=
ps = {Cube, Tetrahedron, Dodecahedron, Icosahedron, Octahedron};
In[20]:=
kmc = Flatten[
  Prepend[MapThread[{#1[#2[[1]]], Sphere[{0, 0, 0}, #2[[2]]]} &, {ps, FoldList[
       With[{length = Solve[Last[Circumsphere[#2[l]]] == #1[[2]]][[1, 1, 2]]}, {length, Last[Insphere[#2[length]]]}] &, {1, 1/2}, Rest[ps]] // FullSimplify}], Sphere[{0, 0, 0}, Sqrt[3]/2]]]
Out[20]=

Kepler’s Mysterium Cosmographicum:

In[21]:=
Manipulate[
 ResourceFunction["OutlinePolygons"][
  Graphics3D[{Opacity[0.5], Take[Reverse@kmc, n]}, Boxed -> False, ClipPlanes -> If[cp, InfinitePlane[{{0, 0, 0}, {1, 1, 0}, {-1, -2, 0}}], None]]], {{n, 6, "zoom out"}, 1, 11, 1}, {{cp, True, "clip plane"}, {True, False}}]
Out[21]=

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