createDurerPolyhedron

CREATEDURERPOLYHEDRON  Create a mesh representing Durer's polyhedron .

   [V, F] = createDurerPolyhedron
   [V, E, F] = createDurerPolyhedron
   Returns a mesh data structure that represents Durer's polyhedron shown
   in "Melancholia". Vertices are stored in V as Nv-by-3 array of 3D
   coordinates, faces are stored in Nf-by-1 cell array containing the
   vertex indices of each face.
   Several hypotheses exist on the exact geometry of the solid. The one
   described in Mathworld (see references) is used here.

   Durer's polyhedron is generated from a centered unit cube. Several
   transforms are applied succesively:
   * Rotation around Oz by PI / 4
   * Rotation around Oy by asec(sqrt(3))
   * z-scaling by sqrt(1 + 3 / sqrt(5) )
   * truncation by two horizontal planes located at a distance of 
       (3 - sqrt(5)) / 2 from each azimutal vertex.

   Durer's polyhedron is composed of six pentagonal faces and 2 triangular
   faces. Pentagonal faces have angles 126, 108, 72, 108, and 126 degrees.
   triangular faces are equilateral.

   Example
     % Display Durer's polyhedron 
     [v f] = createDurerPolyhedron;
     figure; hold on; set(gcf, 'renderer', 'opengl');
     drawMesh(v, f, 'FaceColor', [.7 .7 .7]);
     axis equal; axis([-1 1 -1 1 -1 1]);
     view(3)

   See also
     meshes3d, createCube, createOctahedron

   References
   http://mathworld.wolfram.com/DuerersSolid.html
   http://en.wikipedia.org/wiki/Dürer_graph

0001 function varargout = createDurerPolyhedron(varargin)
0002 %CREATEDURERPOLYHEDRON  Create a mesh representing Durer's polyhedron .
0003 %
0004 %   [V, F] = createDurerPolyhedron
0005 %   [V, E, F] = createDurerPolyhedron
0006 %   Returns a mesh data structure that represents Durer's polyhedron shown
0007 %   in "Melancholia". Vertices are stored in V as Nv-by-3 array of 3D
0008 %   coordinates, faces are stored in Nf-by-1 cell array containing the
0009 %   vertex indices of each face.
0010 %   Several hypotheses exist on the exact geometry of the solid. The one
0011 %   described in Mathworld (see references) is used here.
0012 %
0013 %   Durer's polyhedron is generated from a centered unit cube. Several
0014 %   transforms are applied succesively:
0015 %   * Rotation around Oz by PI / 4
0016 %   * Rotation around Oy by asec(sqrt(3))
0017 %   * z-scaling by sqrt(1 + 3 / sqrt(5) )
0018 %   * truncation by two horizontal planes located at a distance of
0019 %       (3 - sqrt(5)) / 2 from each azimutal vertex.
0020 %
0021 %   Durer's polyhedron is composed of six pentagonal faces and 2 triangular
0022 %   faces. Pentagonal faces have angles 126, 108, 72, 108, and 126 degrees.
0023 %   triangular faces are equilateral.
0024 %
0025 %   Example
0026 %     % Display Durer's polyhedron
0027 %     [v f] = createDurerPolyhedron;
0028 %     figure; hold on; set(gcf, 'renderer', 'opengl');
0029 %     drawMesh(v, f, 'FaceColor', [.7 .7 .7]);
0030 %     axis equal; axis([-1 1 -1 1 -1 1]);
0031 %     view(3)
0032 %
0033 %   See also
0034 %     meshes3d, createCube, createOctahedron
0035 %
0036 %   References
0037 %   http://mathworld.wolfram.com/DuerersSolid.html
0038 %   http://en.wikipedia.org/wiki/Dürer_graph
0039 
0040 % ------
0041 % Author: David Legland
0042 % E-mail: david.legland@inrae.fr
0043 % Created: 2012-07-31, using Matlab 7.9.0.529 (R2009b)
0044 % Copyright 2012-2024 INRA - Cepia Software Platform
0045 
0046 % start from a cube basis
0047 [v, f] = createCube;
0048 
0049 % recenter, rotate, and rescale
0050 v    = v -.5;
0051 rot1 = createRotationOz(pi/4);
0052 rot2 = createRotationOy(asec(sqrt(3)));
0053 sca  = createScaling3d([1 1 sqrt(1+3/sqrt(5))]);
0054 v = transformPoint3d(v, sca * rot2 * rot1);
0055 
0056 % compute the height of the two clipping planes
0057 d = (3 - sqrt(5)) / 2;
0058 zmax = max(v(:,3));
0059 z1 = zmax - d;
0060 
0061 % clip by two horizontal planes
0062 plane1  = createPlane([0 0 z1], [0 0 1]);
0063 [v, f]   = clipConvexPolyhedronByPlane(v, f, plane1);
0064 plane2  = createPlane([0 0 -z1], [0 0 -1]);
0065 [v, f]   = clipConvexPolyhedronByPlane(v, f, plane2);
0066 
0067 % complete with edge information
0068 e = meshEdges(f);
0069 
0070 % format output
0071 varargout = formatMeshOutput(nargout, v, e, f);