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## Copyright (C) 2024 David Legland
## All rights reserved.
##
## Redistribution and use in source and binary forms, with or without
## modification, are permitted provided that the following conditions are met:
##
## 1 Redistributions of source code must retain the above copyright notice,
## this list of conditions and the following disclaimer.
## 2 Redistributions in binary form must reproduce the above copyright
## notice, this list of conditions and the following disclaimer in the
## documentation and/or other materials provided with the distribution.
##
## THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS ''AS IS''
## AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
## IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
## ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE FOR
## ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
## DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
## SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
## CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
## OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
## OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
##
## The views and conclusions contained in the software and documentation are
## those of the authors and should not be interpreted as representing official
## policies, either expressed or implied, of the copyright holders.
function varargout = createDurerPolyhedron(varargin)
%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
% ------
% Author: David Legland
% E-mail: david.legland@inrae.fr
% Created: 2012-07-31, using Matlab 7.9.0.529 (R2009b)
% Copyright 2012-2023 INRA - Cepia Software Platform
% start from a cube basis
[v, f] = createCube;
% recenter, rotate, and rescale
v = v -.5;
rot1 = createRotationOz(pi/4);
rot2 = createRotationOy(asec(sqrt(3)));
sca = createScaling3d([1 1 sqrt(1+3/sqrt(5))]);
v = transformPoint3d(v, sca * rot2 * rot1);
% compute the height of the two clipping planes
d = (3 - sqrt(5)) / 2;
zmax = max(v(:,3));
z1 = zmax - d;
% clip by two horizontal planes
plane1 = createPlane([0 0 z1], [0 0 1]);
[v, f] = clipConvexPolyhedronByPlane(v, f, plane1);
plane2 = createPlane([0 0 -z1], [0 0 -1]);
[v, f] = clipConvexPolyhedronByPlane(v, f, plane2);
% complete with edge information
e = meshEdges(f);
% format output
varargout = formatMeshOutput(nargout, v, e, f);
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