Convex Polygons with Their Insides Chomped Out
- To: mathgroup at smc.vnet.net
- Subject: [mg6566] Convex Polygons with Their Insides Chomped Out
- From: "Seth J. Chandler" <SChandler at uh.edu>
- Date: Tue, 1 Apr 1997 19:01:01 -0500 (EST)
- Organization: University of Houston Law Cen
- Sender: owner-wri-mathgroup at wolfram.com
Does anyone have a good Mathematica algorithm for converting a Polygon
expression (where the polygon has three dimensions and is known to be
convex) into one or more Polygon expressions which, when displayed with a
Graphics3D wrapper, look like the exterior skeleton of the original
polygon. Put differently, I want an attractive "wire frame" of the Polygon
where the "wires" are themselves polygons and not lines.
To give you some idea what I am talking about, here is my inelegant
algorithm which (a) just feels wrong and (b) doesn't work for Polygons with
an odd number of sides. To use this algorithm, write
Polygon[Table[{x[i],y[i],z[i]},{i,1,4}]]/.exterior[0.1]
exterior[z_]:=
RuleDelayed[p_Polygon,
Module[{c,d,i,pts,intsequence,extsequence,thesigns},
pts=First[p];
c=Module[{ppts},ppts=Partition[pts,Floor[Length[pts]/2]];
Last[ppts]-First[ppts]];
thesigns=
Module[{x},x=Floor[Length[pts]/2];
Join[Table[1,{i,1,x}],Table[-1,{i,1,Length[pts]-x}]]];
d=Map[Sqrt[#.#]&,c];
i=MapThread[#1+z*#2*#3*#4&,{pts,Flatten[{c,c},1],Flatten[{d,d},1],
thesigns}];
extsequence=Partition[Append[pts,pts[[1]]],2,1];
intsequence=Reverse/@Partition[Append[i,i[[1]]],2,1];
MapThread[Polygon[Join[#1,#2]]&,{extsequence,intsequence}]
]]