computeDualPolytope

 calculate the dual polytope

 Syntax: dual = computeDualPolytope(polytope)
   or    dual = computeDualPolytope(polytope, rep)

 The dual of a point is a hyperplane and vice versa. In that way the 
   extreme points of a V-representation can be mapped to the halfspaces of
   a dual polytope. For the polytope and the dual polytope there exists a
   one-to-one relationship of extreme points and hyperplanes. Especially
   the following terms are equivalent by duality:
     * a hyperplane is uniquely determined by n linearly independent
       points
     * n linearly independent hyperplanes intersect in a single point

 Usually, if a H-representation is provided, a V-representation is
   returned and vice versa. Using rep, you can specify the output
   representation directly. Use [] or '' to maintain the default behavior.

 The dual mapping is connected to the unit sphere. Hyperplanes that 
   intersect the sphere (do not intersect; touch) are mapped to points 
   outside (inside; on the circumfence) of the sphere and vice versa. For 
   this mapping the following must be considered:
     * no point of a V-representation is equal to the origin. This would
       map to an hyperplane at infinity.
     * no facet distance equals zero. This would map to a point at
       infinity.

 See also: scaleVrep, scaleHrep, normalizeHrep

• convertPolytope converting a polytope between V-representation and H-representation
• identifyPolytope returns basic information on the geometric object
• normalizeHrep normalize length of facet normals to 1

• convertHrepToVrep convert the H-representation of a polytope into V-representation
• convertVrepToHrep convert a V-representation into a H-representation

0001 function dual = computeDualPolytope(poly, rep)
0002 % calculate the dual polytope
0003 %
0004 % Syntax: dual = computeDualPolytope(polytope)
0005 %   or    dual = computeDualPolytope(polytope, rep)
0006 %
0007 % The dual of a point is a hyperplane and vice versa. In that way the
0008 %   extreme points of a V-representation can be mapped to the halfspaces of
0009 %   a dual polytope. For the polytope and the dual polytope there exists a
0010 %   one-to-one relationship of extreme points and hyperplanes. Especially
0011 %   the following terms are equivalent by duality:
0012 %     * a hyperplane is uniquely determined by n linearly independent
0013 %       points
0014 %     * n linearly independent hyperplanes intersect in a single point
0015 %
0016 % Usually, if a H-representation is provided, a V-representation is
0017 %   returned and vice versa. Using rep, you can specify the output
0018 %   representation directly. Use [] or '' to maintain the default behavior.
0019 %
0020 % The dual mapping is connected to the unit sphere. Hyperplanes that
0021 %   intersect the sphere (do not intersect; touch) are mapped to points
0022 %   outside (inside; on the circumfence) of the sphere and vice versa. For
0023 %   this mapping the following must be considered:
0024 %     * no point of a V-representation is equal to the origin. This would
0025 %       map to an hyperplane at infinity.
0026 %     * no facet distance equals zero. This would map to a point at
0027 %       infinity.
0028 %
0029 % See also: scaleVrep, scaleHrep, normalizeHrep
0030 
0031 % The elk-library: convex geometry applied to crystallization modeling.
0032 %   Copyright (C) 2012 Alexander Reinhold
0033 %
0034 % This program is free software: you can redistribute it and/or modify it
0035 %   under the terms of the GNU General Public License as published by the
0036 %   Free Software Foundation, either version 3 of the License, or (at your
0037 %   option) any later version.
0038 %
0039 % This program is distributed in the hope that it will be useful, but
0040 %   WITHOUT ANY WARRANTY; without even the implied warranty of
0041 %   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
0042 %   General Public License for more details.
0043 %
0044 % You should have received a copy of the GNU General Public License along
0045 %   with this program.  If not, see <http://www.gnu.org/licenses/>.
0046 
0047 %% Input
0048 infoPolytope = identifyPolytope(poly);
0049 if ~exist('rep', 'var') || isempty(rep)
0050     switch infoPolytope.rep
0051         case 'Hrep'
0052             rep = 'Vrep';
0053         case 'Vrep'
0054             rep = 'Hrep';
0055     end
0056 end
0057 
0058 switch infoPolytope.rep
0059     case 'Hrep'
0060         for iFacet = 1:infoPolytope.m
0061             poly.A(iFacet,:) = poly.A(iFacet,:) / ...
0062                                 poly.h(iFacet);
0063             poly.h(iFacet) = 1;
0064         end
0065         dual.V = poly.A;
0066     case 'Vrep'
0067         dual.A = poly.V;
0068         dual.h = ones(infoPolytope.m, 1);
0069         dual = normalizeHrep(dual);
0070 end
0071 
0072 dual = convertPolytope(dual, rep);
0073 
0074 end