Re: Mathematica bug in Graphics3D ??

*To*: mathgroup at smc.vnet.net*Subject*: [mg22249] Re: Mathematica bug in Graphics3D ??*From*: Martin Kraus <Martin.Kraus at informatik.uni-stuttgart.de>*Date*: Sat, 19 Feb 2000 01:33:51 -0500 (EST)*Organization*: Institut fuer Informatik, Universitaet Stuttgart*References*: <88juec$9o3@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Axel Kowald wrote: > > Hello, > > I'm using Mathematica 4.0.1.0 under NT4 and when I try: > > Show[Graphics3D[Polygon[{{2.5, Sqrt[3]/2, 0}, {3.5, Sqrt[3]/2, 0}, > {4, Sqrt[3], 0}, {3, 2*Sqrt[3], 0}, {2, 2*Sqrt[3], 0}, > {3, Sqrt[3], 0}} ]], Axes -> True] > > Mathematica draws a polygon with only 5 corners where the last point > {3,Sqrt[3],0} is missing. Am I doing something wrong here or what is > going on ? > > Many thanks > > Axel > Hi Axel, your polygon is non-convex (check it by replacing Polygon by Line) which is a problem for three-dimensional graphics in Mathematica. As explained in the book "Mathematica Graphics - Techniques & Applications" by Tom Wickham-Jones in section 9.3 (Polygon Triangulation) on page 155: "Since there are several methods [to triangulate polygons], Mathematica chooses one that is simple and easy to understand. It joins the first vertex to successive pairs of vertices around the edge." I do not want to discuss this decision. However, it should be noted that this method works consistently for non-flat polygons. Unfortunately, this method does not produce the result you expected for your polygon. In order to get the expected result you should project the polygon into 2d, compute a standard triangulation of the projected polygon and use this triangulation for the 3d points. Tom Wickham-Jones presents some code to do this in his book (included in the ExtendGraphics package in MathSource). Another source of code to do this is my PolygonTriangulation package: http://library.wolfram.com/packages/polygontriangulation/ After installing the package it works like this: << PolygonTriangulation`SimplePolygonTriangulation` Show[TriangulateSimplePolygons3D[ Graphics3D[ Polygon[{{2.5, Sqrt[3]/2, 0}, {3.5, Sqrt[3]/2, 0}, {4, Sqrt[3], 0}, {3, 2*Sqrt[3], 0}, {2, 2*Sqrt[3], 0}, {3, Sqrt[3], 0}}]]], Axes -> True] This will show what you are probably expecting. Greetings Martin Kraus