V_UPOLYHEDRON calculate uniform polyhedron characteristics
Inputs: W Specifies the desired polyhedron in one of three forms:
(a) name e.g. W='cube'; precede by 'dual' for dual or
'laevo' or 'dextro' [default] for chiral polyhedra
(b) index in the list given below e.g. W=6 is the cube; negative for dual
n.{1,2,3,4,5} gives n-sided prism, antiprism, grammic prism, grammic antiprism or grammic crossed antiprism
(c) Wythoff symbol (see below) e.g. W=[3 0 2 4] is the cube
|p q r, p|q r, p q|r or p q r| respectively with 0 for |
using -1 instead of 0 gives the dual of the polyhedron
using -2 (or -3 for dual) gives a reflected version of a snub polyhedron
MD specifies a mode string
'g' plot image
'w' plot wireframe
'f' plot coloured faces
'v' number vertices
't' plot vertex figure (a slice centred on a vertex)
? homogeneous output coordinates
? create net
? segment faces to remove internal portions of the surfaces
? size: [max] diameter=1, [longest] edge=1, include anisotropic normalization
to minimize variance of vertex radius or edge vector length
? orientation: vertex at the top, largest stable base at bottom
Outputs:
VLIST(:,7) gives the [x y z d n e t] for each vertex
x,y,z = position, d=distance from origin, n=valency, e=edge index, t=type (-ve for reflected)
EDGEQ(:,9) has one row for each direction of each edge:
1 v1 first vertex (normally start)
2 v2 second vertex
3 f1 first face (normally on left)
4 f2 second face
5 ev1 next edge around vertex 1 (normally anticlockwise)
6 ef1 next edge around f1 (normally anticlockwise)
7 er reverse edge
8 z twisted edge: clockwise neighbours around v1 and v2 are on the same face
9 sf swap face order: ???
10 sv swap vertex order: v2 preceeds v1 around f1
FLIST(:,7) gives the [x y z d n e t] for each face
x,y,z = unit normal, d=distance from origin, n=valency, e=edge index, t=type (-ve for reflected)
INFO structure containing the following fields:
hemi true if faces are hemispherical (i.e. pass through the origin)
onesided true is one-sided (like a moebius strip)
snub true if a snub polyhedron0001 function [vlist,edgeq,flist,info]=v_upolyhedron(w,md) 0002 %V_UPOLYHEDRON calculate uniform polyhedron characteristics 0003 % 0004 % Inputs: W Specifies the desired polyhedron in one of three forms: 0005 % (a) name e.g. W='cube'; precede by 'dual' for dual or 0006 % 'laevo' or 'dextro' [default] for chiral polyhedra 0007 % (b) index in the list given below e.g. W=6 is the cube; negative for dual 0008 % n.{1,2,3,4,5} gives n-sided prism, antiprism, grammic prism, grammic antiprism or grammic crossed antiprism 0009 % (c) Wythoff symbol (see below) e.g. W=[3 0 2 4] is the cube 0010 % |p q r, p|q r, p q|r or p q r| respectively with 0 for | 0011 % using -1 instead of 0 gives the dual of the polyhedron 0012 % using -2 (or -3 for dual) gives a reflected version of a snub polyhedron 0013 % MD specifies a mode string 0014 % 'g' plot image 0015 % 'w' plot wireframe 0016 % 'f' plot coloured faces 0017 % 'v' number vertices 0018 % 't' plot vertex figure (a slice centred on a vertex) 0019 % 0020 % ? homogeneous output coordinates 0021 % ? create net 0022 % ? segment faces to remove internal portions of the surfaces 0023 % ? size: [max] diameter=1, [longest] edge=1, include anisotropic normalization 0024 % to minimize variance of vertex radius or edge vector length 0025 % ? orientation: vertex at the top, largest stable base at bottom 0026 % 0027 % Outputs: 0028 % VLIST(:,7) gives the [x y z d n e t] for each vertex 0029 % x,y,z = position, d=distance from origin, n=valency, e=edge index, t=type (-ve for reflected) 0030 % EDGEQ(:,9) has one row for each direction of each edge: 0031 % 1 v1 first vertex (normally start) 0032 % 2 v2 second vertex 0033 % 3 f1 first face (normally on left) 0034 % 4 f2 second face 0035 % 5 ev1 next edge around vertex 1 (normally anticlockwise) 0036 % 6 ef1 next edge around f1 (normally anticlockwise) 0037 % 7 er reverse edge 0038 % 8 z twisted edge: clockwise neighbours around v1 and v2 are on the same face 0039 % 9 sf swap face order: ??? 0040 % 10 sv swap vertex order: v2 preceeds v1 around f1 0041 % FLIST(:,7) gives the [x y z d n e t] for each face 0042 % x,y,z = unit normal, d=distance from origin, n=valency, e=edge index, t=type (-ve for reflected) 0043 % INFO structure containing the following fields: 0044 % hemi true if faces are hemispherical (i.e. pass through the origin) 0045 % onesided true is one-sided (like a moebius strip) 0046 % snub true if a snub polyhedron 0047 0048 % This software is based closely on a Mathmatica program described in [1] which, in turn, was based on a 0049 % C program described in [2]. 0050 % 0051 % Wythoff Symbol 0052 % p,q,r define a spherical triangle whose angles at the corners are pi/p, pi/q and pi/r; 0053 % this triangle tiles the sphere if repeatedly reflected in its sides. The 0054 % polyhedron vertices are at the reflections of a seed vertex as follows 0055 % where the Vertex configuration gives the polygon orders of the faces around each vertex: 0056 % |p q r : Vertex at a point such that when rotated around any of p,q,r by twice the angle at that 0057 % corner is displaced by the same distance for each corner. This is a snub polyhedron and 0058 % only even numbers of reflections are used to generate vertices. Configuration {3 p 3 q 3 r} 0059 % p|q r : Vertex at p. Configuration = {q r q r ... q r} with 2n terms where n is the numerator of p 0060 % p q|r : Vertex on pq and on the bisector of angle r. Configuration {p 2r q 2r} 0061 % p q r| : Vertex at incentre: meeting point of all the angle bisectors. Configuration {2p 2q 2r} 0062 % |3/2 5/3 3 5/2 This special case is the great dirhombicosidodecahedron. It is a bit weird because 0063 % many of the edges are shared by four faces instead of the usual two. 0064 % If two of p,q,r = 2 then the third is arbitrary (prisms and antiprisms), otherwise only the numerators 0065 % 1:5 can occur, and 4 and 5 cannot occur together. If all are integers then the poyhedron is convex. 0066 % 0067 % References: 0068 % [1] R. E. Maeder. Uniform polyhedra. The Mathematica Journal, 3 (4): 48-57, 1993. 0069 % [2] Z. Har'El. Uniform solution for uniform polyhedra. Geometriae Dedicata, 47: 57-110, 1993. 0070 % [3] H. S. M. Coxeter, M. S. Longuet-Higgins, and J. C. P. Miller. Uniform polyhedra. 0071 % Philosophical Transactions of the Royal Society A, 246 (916): 401-450, May 1954. 0072 % [4] P. W. Messer. Closed-form expressions for uniform polyhedra and their duals. 0073 % Discrete and Computational Geometry, 27 (3): 353-375, Jan. 2002. 0074 0075 %%%% BUGS and SUGGESTIONS %%%%%% 0076 % (1) we should ensure the "first" edges of the vertices and faces are consistent 0077 % (2) need to sort faces and vertices into a type order 0078 % (3) should ensure that for a non-chiral polyhedron, the vertex polarity alternates 0079 % (4) w=75 does not work 0080 % (5) dual of henispherical poyhedron 0081 % (6) flist not calculated correctly for duals 0082 % (7) add additional stuff into info.* 0083 % (8) vertex figures seem to include additional (duplicated) lines 0084 % (9) sort out when reflected vertices are really rotationally congruent 0085 % (10) could optionally colour by face type 0086 % (11) order alphabetically by noun 0087 % (12) allow abbreviated names + prisms with preceding decimal number 0088 % (13) calculate correct names 0089 % (14) include names for duals 0090 % (15) make "names" etc persistent 0091 0092 % Example slide show: 0093 % for i=1:74, disp(num2str(i)); v_upolyhedron(i); pause(2); end 0094 % for i=5:10, disp(num2str(i)); for j=1:5, v_upolyhedron(i+j/10); pause(2); end, end 0095 0096 % Copyright (C) Mike Brookes 1997 0097 % Version: $Id: v_upolyhedron.m 10865 2018-09-21 17:22:45Z dmb $ 0098 % 0099 % VOICEBOX is a MATLAB toolbox for speech processing. 0100 % Home page: http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/voicebox.html 0101 % 0102 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0103 % This program is free software; you can redistribute it and/or modify 0104 % it under the terms of the GNU General Public License as published by 0105 % the Free Software Foundation; either version 2 of the License, or 0106 % (at your option) any later version. 0107 % 0108 % This program is distributed in the hope that it will be useful, 0109 % but WITHOUT ANY WARRANTY; without even the implied warranty of 0110 % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 0111 % GNU General Public License for more details. 0112 % 0113 % You can obtain a copy of the GNU General Public License from 0114 % http://www.gnu.org/copyleft/gpl.html or by writing to 0115 % Free Software Foundation, Inc.,675 Mass Ave, Cambridge, MA 02139, USA. 0116 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0117 0118 % [line nn] referes to the kaleido Mathematica program on which this is based 0119 % see http://www.mathconsult.ch/showroom/unipoly/unipoly.html#Images 0120 0121 % Variables used 0122 % adjacent adjacency matrix 0123 % chi characteristic 0124 % cosa cos of the angle subtended by a half edge 0125 % d density 0126 % e number of edges 0127 % even number of even faces to remove 0128 % f number of faces 0129 % fi(n) number of faces of type i 0130 % g order of group 0131 % gamma(n) included spherical triangle angle between centre of face, 0132 % vertex and edge for face type i 0133 % hemiQ =1 for hemispherical faces (includes polyhedron centre) 0134 % incidence faces incident at vertex 0135 % k type of group (2=dihedral, 3=tetrahedral, 4=octahedral, 5=icosahedral) 0136 % m number of edges meeting at each vertex (valence) 0137 % mi(n) Number of faces of type i 0138 % n number of face types 0139 % ni(n) number of edges on i'th face type 0140 % p 1st Wythoff number 0141 % q 2nd Wythoff number 0142 % r 3rd Wythoff number 0143 % rot(m) vertex configuration: anti-clockwise list of face types 0144 % around a vertex 0145 % snub(m) =1 for snub triangles 0146 % snubQ =1 for snub polyhedron 0147 % v number of vertices 0148 % vcoord vertex coordinates 0149 % wy(4) Wythoff Symbol 0150 % wyd(4) Wythoff Symbol Denominators 0151 % wyn(4) Wythoff Symbol Numerators 0152 0153 % names: [name abbreviation synonym dual dual-synonym] 0154 persistent names wys prefs 0155 if ~numel(names) 0156 prefs={'dual '; 'dextro '; 'laevo '}; 0157 names={ 0158 'Tetrahedron' 'Tet' '' '' ''; % 1 0159 'truncated tetrahedron' 'Tut' '' '' ''; % 2 0160 'octahemioctahedron' 'Oho' '' '' ''; % 3 0161 'tetrahemihexahedron' 'Thah' '' '' ''; % 4 0162 'Octahedron' 'Oct' '' 'Cube' ''; % 5 0163 'cube' 'Cube' '' 'Octahedron' ''; % 6 0164 'cuboctahedron' 'Co' '' 'rhombic dodecahedron' ''; % 7 0165 'truncated octahedron' 'Toe' '' 'tetrakishexahedron' ''; % 8 0166 'truncated cube' 'Tic' '' 'triakisoctahedron' ''; % 9 0167 'rhombicuboctahedron' 'Sirco' '' 'deltoidal icositetrahedron' ''; % 10 0168 'truncated cuboctahedron' '' '' 'disdyakisdodecahedron' ''; % 11 0169 'snub cube' 'Snic' '' 'pentagonal icositetrahedron' ''; % 12 0170 'small cubicuboctahedron' 'Socco' '' 'small hexacronic icositetrahedron' ''; % 13 0171 'great cubicuboctahedron' 'Gocco' '' 'great hexacronic icositetrahedron' ''; % 14 0172 'cubohemioctahedron' 'Cho' '' 'hexahemioctacron' ''; % 15 0173 'cubitruncated cuboctahedron' 'Cotco' '' 'tetradyakishexahedron' ''; % 16 0174 'great rhombicuboctahedron' 'Girco' '' 'great deltoidal icositetrahedron' ''; % 17 or Querco 0175 'small rhombihexahedron' 'Sroh' '' 'small rhombihexacron' ''; % 18 0176 'stellated truncated hexahedron' 'Quith' '' 'great triakisoctahedron' ''; % 19 0177 'great truncated cuboctahedron' 'Quitco' '' 'great disdyakisdodecahedron' ''; % 20 0178 'great rhombihexahedron' 'Groh' '' 'great rhombihexacron' ''; % 21 0179 'icosahedron' 'Ike' '' 'dodecahedron' ''; % 22 0180 'dodecahedron' 'Doe' '' 'icosahedron' ''; % 23 0181 'icosidodecahedron' 'Id' '' 'rhombic triacontahedron' ''; % 24 0182 'truncated icosahedron' 'Ti' '' 'pentakisdodecahedron' ''; % 25 0183 'truncated dodecahedron' 'Tid' '' 'triakisicosahedron' ''; % 26 0184 'rhombicosidodecahedron' 'Srid' '' 'deltoidal hexecontahedron' ''; % 27 0185 'truncated icosidodechedon' 'Grid' 'great rhombiicosidodecahedron' 'disdyakistriacontahedron' ''; % 28 0186 'snub dodecahedron' 'Snid' '' 'pentagonal hexecontahedron' ''; % 29 0187 'small ditrigonal icosidodecahedron' 'Sidtid' '' 'small triambic icosahedron' ''; % 30 0188 'small icosicosidodecahedron' 'Siid' '' 'small icosacronic hexecontahedron' ''; % 31 0189 'small snub icosicosidodecahedron' 'Seside' '' 'small hexagonal hexecontahedron' ''; % 32 0190 'small dodecicosidodecahedron' 'Saddid' '' 'small dodecacronic hexecontahedron' ''; % 33 0191 'small stellated dodecahedron' 'Sissid' '' 'great dodecahedron' ''; % 34 0192 'great dodecahedron' 'Gad' '' 'small stellated dodecahedron' ''; % 35 0193 'dodecadodecahedron' 'Did' '' 'medial rhombic triacontahedron' ''; % 36 0194 'truncated great dodecahedron' 'Tigid' '' 'small stellapentakisdodecahedron' ''; % 37 0195 'rhombidodecadodecahedron' 'Raded' '' 'medial deltoidal hexecontahedron' ''; % 38 0196 'small rhombidodecahedron' 'Sird' '' 'small rhombidodecacron' ''; % 39 0197 'snub dodecadodecahedron' 'Siddid' '' 'medial pentagonal hexecontahedron' ''; % 40 0198 'ditrigonal dodecadodecahedron' 'Ditdid' '' 'medial triambic icosahedron' ''; % 41 0199 'great ditrigonal dodecicosidodecahedron' 'Gidditdid' '' 'great ditrigonal dodecacronic hexecontahedron' ''; % 42 0200 'small ditrigonal dodecicosidodecahedron' 'Sidditdid' '' 'small ditrigonal dodecacronic hexecontahedron' ''; % 43 0201 'icosidodecadodecahedron' 'Ided' '' 'medial icosacronic hexecontahedron' ''; % 44 0202 'icositruncated dodecadodecahedron' 'Idtid' '' 'tridyakisicosahedron' ''; % 45 0203 'snub icosidodecadodecahedron' 'Sided' '' 'medial hexagonal hexecontahedron' ''; % 46 0204 'great ditrigonal icosidodecahedron' 'Gidtid' '' 'great triambic icosahedron' ''; % 47 0205 'great icosicosidodecahedron' 'Giid' '' 'great icosacronic hexecontahedron' ''; % 48 0206 'small icosihemidodecahedron' 'Seihid' '' 'small icosihemidodecacron' ''; % 49 0207 'small dodecicosahedron' 'Siddy' '' 'small dodecicosacron' ''; % 50 0208 'small dodecahemidodecahedron' 'Sidhid' '' 'small dodecahemidodecacron' ''; % 51 0209 'great stellated dodecahedron' 'Gissid' '' 'great icosahedron' ''; % 52 0210 'great icosahedron' 'Gike' '' 'great stellated dodecahedron' ''; % 53 0211 'great icosidodecahedron' 'Gid' '' 'great rhombic triacontahedron' ''; % 54 0212 'great truncated icosahedron' 'Tiggy' '' 'great stellapentakisdodecahedron' ''; % 55 0213 'rhombicosahedron' 'Ri' '' 'rhombicosacron' ''; % 56 0214 'great snub icosidodecahedron' 'Gosid' '' 'great pentagonal hexecontahedron' ''; % 57 0215 'small stellated truncated dodecahedron' 'Quitsissid' '' 'great pentakisdodekahedron' ''; % 58 0216 'truncated dodecadodecahedron' 'Quitdid' '' 'medial disdyakistriacontahedron' ''; % 59 0217 'inverted snub dodecadodecahedron' 'Isdid' '' 'medial inverted pentagonal hexecontahedron' ''; % 60 0218 'great dodecicosidodecahedron' 'Gaddid' '' 'great dodecacronic hexecontahedron' ''; % 61 0219 'small dodecahemicosahedron' 'Sidhei' '' 'small dodecahemicosacron' ''; % 62 0220 'great dodecicosahedron' 'Giddy' '' 'great dodecicosacron' ''; % 63 0221 'great snub dodecicosidodecahedron' 'Gisdid' '' 'great hexagonal hexecontahedron' ''; % 64 0222 'great dodecahemicosahedron' 'Gidhei' '' 'great dodecahemicosacron' ''; % 65 0223 'great stellated truncated dodecahedron' 'Quitgissid' '' 'great triakisicosahedron' ''; % 66 0224 'great rhombicosidodecahedron' 'Qrid' '' 'great deltoidal hexecontahedron' ''; % 67 or Qrid 0225 'great truncated icosidodecahedron' 'Gaquatid' '' 'great disdyakistriacontahedron' ''; % 68 0226 'great inverted snub icosidodecahedron' 'Gisid' '' 'great inverted pentagonal hexecontahedron' ''; % 69 0227 'great dodecahemidodecahedron' 'Gidhid' '' 'great dodecahemidodecacron' ''; % 70 0228 'great icosihemidodecahedron' 'Geihid' '' 'great icosihemidodecacron' ''; % 71 0229 'small retrosnub icosicosidodecahedron' 'Sirsid' '' 'small hexagrammic hexecontahedron' ''; % 72 0230 'great rhombidodecahedron' 'Gird' '' 'great rhombidodecacron' ''; % 73 0231 'great retrosnub icosidodecahedron' 'Girsid' '' 'great pentagrammic hexecontahedron' ''; % 74 0232 'great dirhombicosidodecahedron' 'Gidrid' '' 'great dirhombicosidodecacron' ''; % 75 0233 'pentagonal prism' 'Pip' '' '' ''; % 76 0234 'pentagonal antiprism' 'Pap' '' '' ''; % 77 0235 'pentagrammic prism' 'Stip' '' '' ''; % 78 0236 'pentagrammic antiprism' 'Stap' '' '' ''; % 79 0237 'pentagrammic crossed antiprism' 'Starp' '' '' ''; % 80 0238 'triangular prism' '' '' '' ''; % 81 0239 'triangular antiprism' '' '' '' ''; % 82 0240 'square prism' '' '' '' ''; % 83 0241 'square antiprism' '' '' '' ''}; % 84 0242 0243 % could multiply by 12 to make integer 0244 wys=[ 0245 2, 3, 2, 3; % 1 'tetrahedron' 'Tet'; 0246 3, 2, 3, 3; % 2 'truncated tetrahedron' 'Tut'; 0247 3, 3/2, 3, 3; % 3 'octahemioctahedron' 'Oho'; 0248 3, 3/2, 3, 2; % 4 'tetrahemihexahedron' 'Thah'; 0249 2, 4, 2, 3; % 5 'octahedron' 'Oct'; 0250 2, 3, 2, 4; % 6 'cube' 'Cube'; 0251 2, 2, 3, 4; % 7 'cuboctahedron' 'Co'; 0252 3, 2, 4, 3; % 8 'truncated octahedron' 'Toe'; 0253 3, 2, 3, 4; % 9 'truncated cube' 'Tic'; 0254 3, 3, 4, 2; % 10 'rhombicuboctahedron' 'Sirco'; 0255 4, 2, 3, 4; % 11 'truncated cuboctahedron' '?'; 0256 1, 2, 3, 4; % 12 'snub cube' 'Snic'; 0257 3, 3/2, 4, 4; % 13 'small cubicuboctahedron' 'Socco'; 0258 3, 3, 4, 4/3; % 14 'great cubicuboctahedron' 'Gocco'; 0259 3, 4/3, 4, 3; % 15 'cubohemioctahedron' 'Cho'; 0260 4, 4/3, 3, 4; % 16 'cubitruncated cuboctahedron' 'Cotco'; 0261 3, 3/2, 4, 2; % 17 'great rhombicuboctahedron' 'Girco?'; 0262 4, 3/2, 2, 4; % 18 'small rhombihexahedron' 'Sroh'; 0263 3, 2, 3, 4/3; % 19 'stellated truncated hexahedron' 'Quith'; 0264 4, 4/3, 2, 3; % 20 'great truncated cuboctahedron' 'Quitco'; 0265 4, 4/3, 3/2, 2; % 21 'great rhombihexahedron' 'Groh'; 0266 2, 5, 2, 3; % 22 'icosahedron' 'Ike'; 0267 2, 3, 2, 5; % 23 'dodecahedron' 'Doe'; 0268 2, 2, 3, 5; % 24 'icosidodecahedron' 'Id'; 0269 3, 2, 5, 3; % 25 'truncated icosahedron' 'Ti'; 0270 3, 2, 3, 5; % 26 'truncated dodecahedron' 'Tid'; 0271 3, 3, 5, 2; % 27 'rhombicosidodecahedron' 'Srid'; 0272 4, 2, 3, 5; % 28 'truncated icosidodecahedron' 'Grid'; or 'great rhombiicosidodecahedron' Grid 0273 1, 2, 3, 5; % 29 'snub dodecahedron' 'Snid'; 0274 2, 3, 5/2, 3; % 30 'small ditrigonal icosidodecahedron' 'Sidtid'; 0275 3, 5/2, 3, 3; % 31 'small icosicosidodecahedron' 'Siid'; 0276 1, 5/2, 3, 3; % 32 'small snub icosicosidodecahedron' 'Seside'; 0277 3, 3/2, 5, 5; % 33 'small dodecicosidodecahedron' 'Saddid'; 0278 2, 5, 2, 5/2; % 34 'small stellated dodecahedron' 'Sissid'; 0279 2, 5/2, 2, 5; % 35 'great dodecahedron' 'Gad'; 0280 2, 2, 5/2, 5; % 36 'dodecadodecahedron' 'Did'; 0281 3, 2, 5/2, 5; % 37 'truncated great dodecahedron' 'Tigid'; 0282 3, 5/2, 5, 2; % 38 'rhombidodecadodecahedron' 'Raded'; 0283 4, 2, 5/2, 5; % 39 'small rhombidodecahedron' 'Sird'; 0284 1, 2, 5/2, 5; % 40 'snub dodecadodecahedron' 'Siddid'; 0285 2, 3, 5/3, 5; % 41 'ditrigonal dodecadodecahedron' 'Ditdid'; 0286 3, 3, 5, 5/3; % 42 'great ditrigonal dodecicosidodecahedron' 'Gidditdid' 0287 3, 5/3, 3, 5; % 43 'small ditrigonal dodecicosidodecahedron' 'Sidditdid' 0288 3, 5/3, 5, 3; % 44 'icosidodecadodecahedron' 'Ided'; 0289 4, 5/3, 3, 5; % 45 'icositruncated dodecadodecahedron' 'Idtid'; 0290 1, 5/3, 3, 5; % 46 'snub icosidodecadodecahedron' 'Sided'; 0291 2, 3/2, 3, 5; % 47 'great ditrigonal icosidodecahedron' 'Gidtid'; 0292 3, 3/2, 5, 3; % 48 'great icosicosidodecahedron' 'Giid'; 0293 3, 3/2, 3, 5; % 49 'small icosihemidodecahedron' 'Seihid'; 0294 4, 3/2, 3, 5; % 50 'small dodecicosahedron' 'Siddy'; 0295 3, 5/4, 5, 5; % 51 'small dodecahemidodecahedron' 'Sidhid'; 0296 2, 3, 2, 5/2; % 52 'great stellated dodecahedron' 'Gissid'; 0297 2, 5/2, 2, 3; % 53 'great icosahedron' 'Gike'; 0298 2, 2, 5/2, 3; % 54 'great icosidodecahedron' 'Gid'; 0299 3, 2, 5/2, 3; % 55 'great truncated icosahedron' 'Tiggy'; 0300 4, 2, 5/2, 3; % 56 'rhombicosahedron' 'Ri'; 0301 1, 2, 5/2, 3; % 57 'great snub icosidodecahedron' 'Gosid'; 0302 3, 2, 5, 5/3; % 58 'small stellated truncated dodecahedron' 'Quitsissid' 0303 4, 5/3, 2, 5; % 59 'truncated dodecadodecahedron' 'Quitdid'; 0304 1, 5/3, 2, 5; % 60 'inverted snub dodecadodecahedron' 'Isdid'; 0305 3, 5/2, 3, 5/3; % 61 'great dodecicosidodecahedron' 'Gaddid'; 0306 3, 5/3, 5/2, 3; % 62 'small dodecahemicosahedron' 'Sidhei'; 0307 4, 5/3, 5/2, 3; % 63 'great dodecicosahedron' 'Giddy'; 0308 1, 5/3, 5/2, 3; % 64 'great snub dodecicosidodecahedron' 'Gisdid'; 0309 3, 5/4, 5, 3; % 65 'great dodecahemicosahedron' 'Gidhei'; 0310 3, 2, 3, 5/3; % 66 'great stellated truncated dodecahedron' 'Quitgissid' 0311 3, 5/3, 3, 2; % 67 'quasirhombicosidodecahedron' 'Qrid'; 'great rhombicosidodecahedron' 'Nonconvex great rhombicosidodecahedron 0312 4, 5/3, 2, 3; % 68 'great truncated icosidodecahedron' 'Gaquatid'; 0313 1, 5/3, 2, 3; % 69 'great inverted snub icosidodecahedron' 'Gisid'; 0314 3, 5/3, 5/2, 5/3; % 70 'great dodecahemidodecahedron' 'Gidhid'; 0315 3, 3/2, 3, 5/3; % 71 'great icosihemidodecahedron' 'Geihid'; 0316 1, 3/2, 3/2, 5/2; % 72 'small retrosnub icosicosidodecahedron' 'Sirsid'; 0317 4, 3/2, 5/3, 2; % 73 'great rhombidodecahedron' 'Gird'; 0318 1, 3/2, 5/3, 2; % 74 'great retrosnub icosidodecahedron' 'Girsid'; 0319 5, 3/2, 5/3, 3; % 75 'great dirhombicosidodecahedron' 'Gidrid'; 0320 3, 2, 5, 2; % 76 'pentagonal prism' 'Pip'; 0321 1, 2, 2, 5; % 77 'pentagonal antiprism' 'Pap'; 0322 3, 2, 5/2, 2; % 78 'pentagrammic prism' 'Stip'; 0323 1, 2, 2, 5/2; % 79 'pentagrammic antiprism' 'Stap'; 0324 1, 2, 2, 5/3; % 80 'pentagrammic crossed antiprism' 'Starp' 0325 3, 2, 3, 2; % 81 'triangular prism' ; 0326 1, 2, 2, 3; % 82 'triangular antiprism' ; 0327 3, 2, 4, 2; % 83 'square prism' ; 0328 1, 2, 2, 4]; % 84 'square antiprism' ; 0329 end 0330 0331 if nargin<2 0332 md=''; 0333 end 0334 dual=0; % dual polyhedron 0335 dextro=0; % reflected (for snub polyhedra only) 0336 if ischar(w) 0337 i=0; 0338 while i<length(prefs) 0339 i=i+1; 0340 if length(w)>length(prefs{i}) && strcmpi(w(1:length(prefs{i})),prefs{i}) 0341 switch i 0342 case 1 0343 dual=1; 0344 case 2 0345 dextro=1; 0346 end 0347 w=w(length(prefs{i})+1:end); 0348 i=0; 0349 end 0350 end 0351 0352 wyidx=find(strcmpi(w,names(:)),1); 0353 if isempty(wyidx) 0354 % check for prism names here 0355 error('Cannot find %s',w); 0356 end 0357 if wyidx>3*size(names,1) 0358 dual=1-dual; 0359 end 0360 wyidx=1+rem(wyidx-1,size(names,1)); 0361 wy=wys(wyidx,:); 0362 else 0363 if length(w)==1 0364 dual=w<0; 0365 wyidx=abs(w); 0366 wyjdx=floor(wyidx); 0367 wykdx=round(10*(wyidx-wyjdx)); 0368 switch wykdx 0369 case 1 % prism 0370 wy=[3 2 wyjdx 2]; 0371 case 2 % antiprism 0372 wy=[1 2 2 wyjdx]; 0373 case 3 % grammic prism 0374 wy=[3 2 wyjdx/2 2]; 0375 case 4 % grammic antiprism 0376 wy=[1 2 2 wyjdx/2]; 0377 case 5 % grammic crossed antiprism 0378 wy=[1 2 2 wyjdx/3]; 0379 otherwise 0380 if wyjdx>=1 && wyjdx<=size(wys,1) 0381 wy=wys(wyjdx,:); 0382 else 0383 error('Polyhedron number out of range'); 0384 end 0385 end 0386 elseif length(w)==4 0387 vbar=find(w<=0,1); 0388 if ~numel(vbar) 0389 error('Invalid Wythoff symbol: %g %g %g %g %g',w); 0390 end 0391 dual=mod(w(vbar),2); % least significant bit indicates dual 0392 dextro=mod(1+w(vbar),4)>=2; % second bit indicates reflected version 0393 wy=[vbar w(1:vbar-1) w(vbar+1:4)]; 0394 [xx,wyidx]=min(sum((wys-repmat(wy,size(wys,1),1)).^2,2)); 0395 if abs(xx)>1e-3 0396 if sum(wy==2)<2 0397 error('Invalid Wythoff symbol: %g %g %g %g %g',w); 0398 else 0399 % need to sort out prism names here 0400 end 0401 end 0402 elseif length(w)==5 0403 if w(1)<=0 && all(round(12*w(2:end))==[18 40 36 30]) % [3/2 5/3 3 5/2] 0404 dual=w(1)<0; 0405 wyidx=75; 0406 wy=wys(wyidx,:); 0407 else 0408 error('Invalid Wythoff symbol: %g %g %g %g %g',w); 0409 end 0410 else 0411 error('Invalid polyhedron specification'); 0412 end 0413 end 0414 0415 0416 [wyn,wyd]=rat(wy(2:4)); % convert to rational numbers 0417 wy(2:4)=wyn./wyd; % force exact rational values 0418 p=wy(2); 0419 q=wy(3); 0420 r=wy(4); 0421 hemiQ = 0; % includes hemispherical faces that go through polyhedron centre [ p q | r] with pq=p+q 0422 onesidedQ = 0; % one-sided polyhedron (aka: non-orientable) [ p q r |] with exactly one of p,q,r having an even denominator 0423 evenQ = 0; % identifies which of p, q, r has an even denominator 0424 even=1; 0425 snubQ = 0; % snub polyhedron: [ | p q r ] 0426 allrot = 1; % all vertices a congruent with rotations (no reflections needed) 0427 0428 % call to AnalyseWythoff[line 105] 0429 0430 k= max(wyn); 0431 if sum(wy(2:4)==2)>=2 % check if it is a prism 0432 % this is a prism - treat specially [note mathematica line 86 has redundant chack for 2 >5] 0433 g=4*k; % order of group 0434 k=2; 0435 else % not a prism: only numerators 1,2,3,4,5 are allowed 0436 if (any(wy(2:4)<=1) || (k>5) || any(wyn==4) && any(wyn==5)) 0437 error('Invalid Wythoff numbers [%d/%d %d/%d %d/%d]',[wyn;wyd]); 0438 end 0439 g=24*k/(6-k); % order of group 0440 end 0441 if abs(wy(1))==5 % special case 0442 chi = 0; 0443 d = 0; 0444 else 0445 chi = (sum(wyn.^(-1))-1)*g/2; 0446 d = (sum(wy(2:4).^(-1))-1)*g/4; 0447 if d<0 0448 error('density < 0'); 0449 end 0450 end 0451 if abs(wy(1))==5 0452 error('great dirhombicosidodecahedron not implemented'); 0453 end 0454 switch abs(wy(1)) 0455 case {1,5} % [ | p q r ] snub polyhedron 0456 n=4; % number of face types 0457 m=6; % valence of vertices 0458 v=g/2; % number of vertices 0459 % ni=[3 wy(2:4)]; % type of each face 0460 % mi=[3 1 1 1]; % number of faces of each type 0461 nimi=[3 wy(2:4); 3 1 1 1]'; 0462 % rot = [1 2+dextro 1 3-dextro 1 4]; 0463 snubQ=1; 0464 % snub=[1 0 1 0 1 0]; 0465 rotsnub=[1 2+dextro 1 3-dextro 1 4; repmat([1 0],1,3)]'; 0466 case 2 % [ p | q r ] 0467 n=2; 0468 m=2*wyn(1); 0469 v=g/m; 0470 % ni=wy(3:4); 0471 % mi=[p p]; % these face counts may be fractional: m counts their numerators 0472 % we could ill in the numerator denominator columns, nimi(:,3:4) here instead of later 0473 nimi=[wy(3:4); p p]'; 0474 % rot = repmat(1:2,1,wyn(1)); 0475 rotsnub = repmat([1 0; 2 0],wyn(1),1); 0476 case 3 % [ p q | r ] 0477 n=3; 0478 m=4; 0479 v=g/2; 0480 % ni=[2*r wy(2:3)]; 0481 % mi = [2 1 1]; 0482 nimi=[2*r wy(2:3); 2 1 1]'; 0483 % rot = [1 2 1 3]; 0484 rotsnub = [1 2 1 3; zeros(1,4)]'; 0485 if abs(p-q/(q-1))<1e-6 0486 hemiQ=1; 0487 d=0; 0488 if (p~=2 && ~(wy(4)==3 && any(wy(2:3)==3))) 0489 onesidedQ=1; 0490 v=v/2; 0491 chi=chi/2; 0492 end 0493 end 0494 case 4 % [ p q r | ] 0495 n=3; 0496 m=3; 0497 v=g; 0498 % ni=2*wy(2:4); 0499 % mi=ones(1,3); 0500 nimi=[2*wy(2:4); ones(1,3)]'; 0501 % rot=1:3; 0502 rotsnub=[1:3; zeros(1,3)]'; 0503 allrot=(wy(2)==wy(3)) || (wy(2)==wy(4)) || (wy(3)==wy(4)); % isosceles so all 0504 evenden=find(~rem(wyd,2)); % check for even denominators 0505 if (length(evenden)>1) 0506 error('Multiple even denominators are not allowed'); 0507 end 0508 if ~isempty(evenden) 0509 even = evenden; 0510 evenQ=1; 0511 onesidedQ = 1; 0512 d=0; 0513 v=v/2; 0514 chi = chi - g/wyn(even)/2; % check this ***** 0515 end 0516 otherwise 0517 error('Invalid polyhedron type %d',wy(1)); 0518 end 0519 % [line 155] call sortAndMerge [sortAndMerge: line 220] 0520 % save stuff to prevent overwriting 0521 n_1=n; 0522 m_1=m; 0523 % now do sort merge based on rotsnub and nimi 0524 % rotsnub(:,2) = [rot snub] 0525 % nimi(:,4) = [ni mi mi-num mi-den] 0526 0527 nimi(nimi(:,1)==2,1)=-2; % make equal to -2 to force to bottom of list 0528 [nimi,inim]=sortrows(nimi,-1); 0529 jnim=zeros(n,1); 0530 jnim(inim)=1:n; 0531 msk=[~0; nimi(2:end,1)<nimi(1:end-1,1)]; % identify distinct values 0532 cmsk=cumsum(msk); % destination of each value 0533 nimi(msk,2)=sparse(1,cmsk,nimi(:,2)); % add up repreated counts 0534 nimi=nimi(msk,:); % select just the distinct values 0535 % cmsk(jnim) maps original to new positions 0536 [nimi(:,3),nimi(:,4)]=rat(nimi(:,2)); 0537 rotsnub(:,1)=cmsk(jnim(rotsnub(:,1))); 0538 even=cmsk(jnim(even)); 0539 if nimi(end,1)<0 % remove digons 0540 if size(nimi,1)==1 0541 error('Degenerate polyhedron (digons only)'); 0542 end 0543 m=m-nimi(end,3); % note that m=sum(nimi(:,3)) not sum(nimi(:,2)) as you might expect 0544 nimi(end,:)=[]; 0545 n=size(nimi,1); 0546 rotsnub(rotsnub(:,1)>n,:)=[]; % abolish references to digons 0547 end 0548 0549 % original sort and merge 0550 0551 % [ss,tr]=sort(-ni); % sort into descending order 0552 % itr=zeros(1,n); 0553 % itr(tr)=1:n; 0554 % ni=ni(tr); 0555 % mi=mi(tr); 0556 % rot = itr(rot); 0557 % if evenQ 0558 % even = itr(even); 0559 % end 0560 % % now merge equal faces 0561 % i=1; 0562 % while i<n 0563 % if ni(i)==ni(i+1) 0564 % mi(i)=mi(i)+mi(i+1); 0565 % mi(i+1)=[]; 0566 % ni(i+1)=[]; 0567 % n=n-1; 0568 % even=even-(even>i); 0569 % rot = rot - (rot>i); 0570 % else 0571 % i=i+1; 0572 % end 0573 % end 0574 % [mii,mij]=rat(mi); % find numerator 0575 % i=find(ni==2); % find digons 0576 % if ~isempty(i) 0577 % if n==1 0578 % error('Degenerate (digons only)'); 0579 % end 0580 % i=i(1); 0581 % m=m-mii(i); % reduce total valance 0582 % ni(i)=[]; 0583 % mi(i)=[]; 0584 % mii(i)=[]; 0585 % even=even-(even>i); 0586 % if snubQ 0587 % snub(rot==i)=[]; 0588 % end 0589 % rot(rot==i)=[]; 0590 % rot = rot - (rot>i); 0591 % n=n-1; 0592 % end 0593 % 0594 % % check that new version is correct 0595 % 0596 % 0597 % if any(nimi(:,1:3)~=[ni' mi' mii']) 0598 % error('nimi has errors'); 0599 % end 0600 % if snubQ 0601 % if any(rotsnub~=[rot' snub']) 0602 % error('rotsnub has errors'); 0603 % end 0604 % else 0605 % if any(rotsnub(:,1)~=rot') 0606 % error('rotsnub has errors'); 0607 % end 0608 % end 0609 % if evenQ && even~=even_1 0610 % error('even has errors'); 0611 % end 0612 0613 % [line 159] Solve fundamental equations [line 266] 0614 % doesn't converge well for a dodecahedron 0615 % we could avoid the iteration when n=1 0616 % 0617 % We want to solve the following set of equations for cosa and gammai: 0618 % (a) sum(mi.*gammai)=pi 0619 % (b) cos(alphai) = cosa*sin(gammai) 0620 % where ni, mi, alphai and gammai are vectors of length m, the number of edges at a vertex 0621 % ni gives the number of sides in each face type in decreasing order (possibly fractional) 0622 % mi is the repretition count of each face type (possibly fractional) 0623 % alphai=pi * ni.^(-1) is the angle subtended by a half edge at the centre of a face 0624 % gammai (> pi/2 - alphai) is half the spherical angle subtended by a face at the vertex 0625 % cosa (< 1) is the distance of the edge centre from the origin if the radius of the vertex is unity 0626 % we solve this iteratively using gammai(1) as the independent variable 0627 0628 0629 alphai = pi*nimi(:,1).^(-1); % half of external angle of polygon 0630 calphai = cos(alphai); 0631 ca1 = calphai(1); 0632 calphai(1)=[]; 0633 % initial values 0634 gammai = pi/2 - alphai; % half of internal angle of polygon 0635 delta = pi - nimi(:,2)'*gammai; 0636 % we could initialize it to be exact for a single face type 0637 % g1n=gammai(1)*pi/(mi*gammai') 0638 % cosa = ca1/sin(g1n); 0639 % gammai = [g1n asin(calphai/cosa)]; 0640 % delta = pi - mi*gammai'; 0641 % should check here to see if delta is better than before and g1n is valid 0642 iter=0; 0643 while (abs(delta) > 1e-10) 0644 g1n = gammai(1) + delta*tan(gammai(1))/(nimi(:,2)'*tan(gammai)); 0645 if (g1n<0 || g1n>pi) 0646 error ('Gamma out of range'); 0647 end 0648 cosa = ca1/sin(g1n); 0649 gammai = [g1n; asin(calphai/cosa)]; 0650 delta = pi - nimi(:,2)'*gammai; 0651 iter=iter+1; 0652 end 0653 gamma = gammai; 0654 cosa = ca1/sin(gammai(1)); 0655 0656 % [line 165] postprocess special cases 0657 0658 if evenQ 0659 nimi(even,:)=[]; 0660 gamma(even)=[]; 0661 nh=n-1; 0662 n=2*nh; 0663 m=4; 0664 % ni=[ni 1+fliplr(ni-1).^(-1)]; 0665 nimi=repmat(nimi,2,1); 0666 nimi(nh+1:end,1)=1+(nimi(nh:-1:1,1)-1).^(-1); 0667 gamma=[gamma; -flipud(gamma)]; 0668 % mi=repmat(3-nh,1,n); 0669 % mii=mi; 0670 nimi(:,2:3)=repmat(3-nh,n,2); 0671 rotsnub=[1 nh n 1+nh; zeros(1,4)]'; 0672 end 0673 if wy(1)==5 0674 error('not yet implemented'); 0675 % needs to use rotsnub and nimi 0676 n=5; 0677 m=8; 0678 hemiQ = 1; 0679 d=0; 0680 mi=[4 1 1 1 1]; 0681 mii=mi; 0682 ni=[4 ni(1:3) ni(1)/(ni(1)-1)]; 0683 gamma=[pi/2 gamma(1:3) -gamma(1)]; 0684 rot=[rot+[0 1 0 1 0 1] 1 5]; % [1 4 1 3 1 2 1 5] 0685 snub=[snub 1 0]; 0686 end 0687 0688 0689 0690 % [line 197] count vertices and faces [count: line 286] 0691 0692 e = m*v/2; % number of edges = valency * vertices /2 0693 [nii,nij]=rat(nimi(:,1)); % find numerators 0694 fi = v*nimi(:,3)./nii; 0695 f = sum(fi); 0696 if (d>0) && (gamma(1)> pi/2) % [might be better to use cycles rather than radians for gamma] 0697 d = fi(1)-d; 0698 end 0699 if wy(1)==5 0700 chi = v - e + f; % Euler characteristic 0701 end 0702 0703 % [line 199] generate vertex coordinates and adjacency matrix [vertices: line 297] 0704 % we currently scale it so the edge mid-points have unit radius 0705 0706 adj=zeros(v,m); % j'th vertex adjacent to vertex i anti-clockwise; 0707 frot = zeros(1,v); 0708 rev = zeros(v,1); 0709 vlist = zeros(v+1,7); % vertex information: [x y z d n e t] for each vertex 0710 % x,y,z = position, d=distance from origin, n=valency, e=edge index, t=type (-ve for reflected) 0711 vlist(:,5)=m; % all vertices have the same valency 0712 v1 = [0 0 1]/cosa; % first vertex 0713 frot(1) = 1; 0714 adj(1,1) = 2; 0715 v2 = [2*cosa*sqrt(1-cosa^2), 0, 2*cosa^2-1]/cosa; % second vertex 0716 if snubQ 0717 frot(2) = m*(1-rotsnub(m,2))+rotsnub(m,2); % inefficient 0718 adj(2,1)=1; 0719 else % reflexible 0720 frot(2) = 1; 0721 rev(2) = 1; 0722 adj(2,m)=1; 0723 end 0724 vlist(1,1:3)=v1; 0725 vlist(2,1:3)=v2; 0726 nv = 2; 0727 i = 1; 0728 skew=zeros(3,3); 0729 skewp=[6 7 2]; % index of positive entries 0730 skewn=[8 3 4]; % index of negative entries 0731 cosg=cos(2*gamma); 0732 sing=sin(2*gamma); 0733 eye3=eye(3); 0734 % veqth=cos(acos(cosa)/(d+1)); % threshold for vertex equality test = cosa * |vlist(1,1:3)|^2 0735 veqth=0.999999/(cosa^2); % threshold for vertex equality test = cosa * |vlist(1,1:3)|^2 0736 while i<=nv % loop for each vertex 0737 if rev(i) 0738 one = -1; 0739 start = m-1; 0740 limit = 1; 0741 else 0742 one = 1; 0743 start = 2; 0744 limit = m; 0745 end 0746 k = frot(i); % v_rotation to use first 0747 v1 = vlist(i,1:3); % the centre of v_rotation 0748 v1 = v1/sqrt(v1*v1'); % normalize to unit length [not clear why we don't make unit length in the first place] 0749 v2 = vlist(adj(i,start-one),1:3); 0750 for j=start:one:limit 0751 % R=wwT+cos(x)(I-wwT)+sin(x)SKEW(w) 0752 % SKEW(a) = [0 -a3 a2; a3 0 -a1; -a2 a1 0] 0753 skew(skewp)=v1; 0754 skew(skewn)=-v1; 0755 rsym=v1'*v1; 0756 rotk=rotsnub(k,1); 0757 rotmat=rsym+cosg(rotk)*(eye3-rsym)-one*sing(rotk)*skew; 0758 v2=v2*rotmat; % rotate v2 poition around v1 0759 vlist(nv+1,1:3)=v2; % add into list in case it is good 0760 nvi=find(vlist(:,1:3)*v2'>veqth,1); % take the first matching vertex 0761 % cmatch=1-vlist(nvi,:)*v2'*cosa^2 % diagnostic printout for vertex match test (0 for perfect match) 0762 adj(i,j)=nvi; % save as next vertex 0763 lastk = k; 0764 k=1+mod(k,m); % increment k circularly in the range 1:m 0765 if nvi>nv % we have a new vertex 0766 if snubQ % if a snub polyhedron 0767 % Mathematica: frot[[nvi]] = If[ !sn[[lastk]], lastk, If[ !sn[[k]], next[k, m], k ] ]; 0768 % A snub triangle plays the role of the next available snub triangle for the next vertex 0769 frot(nvi)=lastk+rotsnub(lastk,2)*(rotsnub(k,2)*k-lastk+(1-rotsnub(k,2))*(1+mod(k,m))); 0770 rev(nvi)=0; % snub polyhedra always have rev=0 0771 adj(nvi,1)=i; 0772 else 0773 frot(nvi)=k; 0774 rev(nvi)=(1+one)/2; % = 1-rev(i) 0775 adj(nvi,1+(m-1)*rev(nvi))=i; 0776 end 0777 nv=nvi; 0778 if nv>v 0779 error('Too many vertices found'); 0780 end 0781 end 0782 end 0783 i=i+1; 0784 end 0785 if (nv~=v) 0786 error('Not all vertices found'); 0787 end 0788 vlist(v+1,:)=[]; % remove the dummy extra vertex 0789 vlist(:,7)=1-2*rev; % vertex types all all +1 or -1 0790 0791 % construct edge map 0792 % edgeq: 1=v1 2=v2 3=f1 4=f2 5=ev1 6=ef1 7=er 8=z 9=sf 10=sv] 0793 % 1 v1 first vertex (normally start) 0794 % 2 v2 second vertex 0795 % 3 f1 first face (normally on left) 0796 % 4 f2 second face 0797 % 5 ev1 next edge around vertex 1 (normally anticlockwise) 0798 % 6 ef1 next edge around f1 (normally anticlockwise) 0799 % 7 er reverse edge 0800 % 8 z twisted edge: clockwise neighbours around v1 and v2 are on the same face 0801 % 9 sf swap face order: ??? 0802 % 10 sv swap vertex order: v2 preceeds v1 around f1 0803 0804 edgeq=zeros(v*m,10); % OLD *** ===[reverse_edge start_vertex left_face next_edge_at_vertex next_edge_on_face] 0805 edgeq(:,1:2)=[repmat((1:v)',m,1) adj(:)]; % 2=v_start 3=v_end 0806 edgeq(:,3:4)=edgeq(:,1:2)*[v 1; 1 v]; % encode start and end as a single integer 0807 [xx,ia]=sort(edgeq(:,3)); 0808 [xx,ib]=sort(edgeq(:,4)); 0809 edgeq(:,3:4)=0; 0810 ib(ib)=1:v*m; % make reverse index 0811 edgeq(:,7)=ia(ib); % index to reverse direction edge: 1=e_reverse 0812 edgeq(:,5)=[v+1:v*m 1:v]; % adjacent edge anti-clockwise around vertex: 6=e_next_at_start 0813 ia(edgeq(:,5))=1:v*m; % adjacent edge clockwise around vertex; alternatively ia = mod((1:v*m)-v,v*m)' 0814 edgeq(:,6)=ia(edgeq(:,7)); % adjacent edge anti-clockwise around face [but not necessarily correct 0815 fpt=zeros(f,2); % edge count and pointer to an edge on a face 0816 iff=0; % face index 0817 if onesidedQ 0818 edgeq(:,8)=reshape(mod(repmat(frot',1,m)+repmat((1:m),v,1),2),v*m,1); %face type modulo 2 0819 edgeq(:,8)=abs(edgeq(:,8)-edgeq(edgeq(:,6),8)); % 1 = twisted edge 0820 while iff<f 0821 iee=find(edgeq(:,3)==0,1); % find an edge without a left face 0822 if ~numel(iee) 0823 error('Not enough faces found'); 0824 end 0825 iff=iff+1; 0826 fpt(iff,2)=iee; 0827 nee=0; 0828 flip=abs((edgeq(iee,6)==edgeq(edgeq(iee,7),5))-edgeq(iee,8)); % check if this initial edge is flipped on output 0829 while edgeq(iee,3)~=iff 0830 if edgeq(iee,3) % use the reverse edge if this one is already taken 0831 iee=edgeq(iee,7); 0832 edgeq(pee,6)=iee; % correct the previous exit edge taken 0833 edgeq(iee,10)=1; % use it in the reverse direction 0834 edgeq(iee,6)=ia(iee); % normal exit for this direction 0835 jee=edgeq(iee,5); % alternate exit edge if flipping 0836 else 0837 if flip && ~edgeq(iee,4) % we are entering on the usual reverse exit 0838 edgeq(edgeq(iee,7),6)=edgeq(iee,5); % So give the reverse exit a valid path 0839 end 0840 jee=edgeq(edgeq(iee,7),5); % alternate exit edge if flipping 0841 end 0842 if edgeq(iee,3) 0843 error('Edge already in use'); 0844 end 0845 edgeq(iee,3)=iff; 0846 edgeq(edgeq(iee,7),4)=iff; % mark right face in reverse edge 0847 nee=nee+1; 0848 flip=abs(flip-edgeq(iee,8)); 0849 if flip % use alternate edge 0850 edgeq(iee,6)=jee; % mark actual edge used 0851 end 0852 pee=iee; % save previous iee value 0853 iee=edgeq(iee,6); % step around the face 0854 end 0855 fpt(iff,1)=nee; 0856 end 0857 else % a normal two-sided polygon 0858 while iff<f 0859 iee=find(edgeq(:,3)==0,1); % find an edge without a left face 0860 iff=iff+1; 0861 fpt(iff,2)=iee; 0862 nee=0; 0863 while edgeq(iee,3)~=iff 0864 if edgeq(iee,3) 0865 error('Edge already in use'); 0866 end 0867 edgeq(iee,3)=iff; 0868 edgeq(edgeq(iee,7),4)=iff; % mark right face in reverse edge 0869 nee=nee+1; 0870 iee=edgeq(iee,6); % step around the face 0871 end 0872 fpt(iff,1)=nee; 0873 end 0874 end 0875 % [(1:v*m)' edgeq] 0876 0877 vlist(:,4)=sqrt(sum(vlist(:,1:3).^2,2)); % fill in the radii 0878 vlist(:,6)=(1:v)'; 0879 % now fill in the flist array 0880 % [x y z d n e t] for each face 0881 flist=zeros(f,7); 0882 [ia,ib]=sort(edgeq(:,3)); % finst all the edges belonging to each face 0883 ix=[1; 1+find(ia(2:end)>ia(1:end-1)); 2*e+1]; 0884 flist(:,6)=ib(ix(1:f)); % pointer to first edge for each face 0885 flist(:,5)=ix(2:end)-ix(1:end-1); % size of each face 0886 % edgeq: 1=v1 2=v2 3=f1 4=f2 5=ev1 6=ef1 7=er 8=z 9=sf 10=sv] 0887 tedge=flist(:,6); % this edge 0888 nedge=edgeq(tedge,6); % next edge 0889 vmid=vlist(edgeq(tedge+2*e*(1-edgeq(tedge,10))),1:3); 0890 flist(:,1:3)=cross(vlist(edgeq(nedge+2*e*(1-edgeq(nedge,10))),1:3)-vmid,vlist(edgeq(tedge+2*e*edgeq(tedge,10)),1:3)-vmid,2); 0891 flist(:,1:3)=flist(:,1:3)./repmat(sqrt(sum(flist(:,1:3).^2,2)),1,3); 0892 flist(:,4)=sum(flist(:,1:3).*vmid,2); 0893 0894 % now check for dual 0895 0896 if dual 0897 vlisto=vlist; % save vlist 0898 vlist=flist; % old faces are new vertices 0899 flist=vlisto; 0900 % edgeq: 1=v1 2=v2 3=f1 4=f2 5=ev1 6=ef1 7=er 8=z 9=sf 10=sv] 0901 edgeq=edgeq(:,[3 4 2 1 6 5 7 8 10 9]); % swap vertices and faces in the edge list 0902 erev=edgeq(:,7); % reverse edge 0903 edgeq(:,6)=erev(edgeq(erev,6)); 0904 flist(:,6)=erev(flist(:,6)); 0905 if hemiQ % cannot invert through centre 0906 error('Cannot take dual'); 0907 else 0908 vlist(:,4)=vlist(:,4).^(-1); 0909 end 0910 vlist(:,1:3)=vlist(:,1:3).*repmat(vlist(:,4),1,3); 0911 v=size(vlist,1); 0912 vlist(:,1:3)=vlist(:,1:3)-repmat(mean(vlist(:,1:3),1),v,1); 0913 f=size(flist,1); % recalculate face positions from scratch 0914 tedge=flist(:,6); % this edge 0915 nedge=edgeq(tedge,6); % next edge 0916 vmid=vlist(edgeq(tedge+2*e*(1-edgeq(tedge,10))),1:3); 0917 flist(:,1:3)=cross(vlist(edgeq(nedge+2*e*(1-edgeq(nedge,10))),1:3)-vmid,vlist(edgeq(tedge+2*e*edgeq(tedge,10)),1:3)-vmid,2); 0918 flist(:,1:3)=flist(:,1:3)./repmat(sqrt(sum(flist(:,1:3).^2,2)),1,3); 0919 flist(:,4)=sum(flist(:,1:3).*vmid,2); 0920 end 0921 0922 info.snub=snubQ>0; 0923 info.onesided=onesidedQ>0; 0924 info.hemi=hemiQ>0; 0925 0926 % create Wythoff string 0927 wystr=''; 0928 j=0; 0929 for i=1:4 0930 if i==abs(wy(1)) 0931 wystr=[wystr '| ']; 0932 j=1; 0933 else 0934 if (wyd(i-j)==1) 0935 wystr=[wystr num2str(wyn(i-j)) ' ']; 0936 else 0937 wystr=[wystr num2str(wyn(i-j)) '/' num2str(wyd(i-j)) ' ']; 0938 end 0939 end 0940 end 0941 info.wythoff=wystr(1:end-1); 0942 info.vef=[v e f]; 0943 info.chi=chi; % Euler Characteristic 0944 0945 % now draw an image if requested 0946 0947 if ~nargout || any(md=='g') 0948 clf; 0949 if any(md=='t') 0950 [nii,nij]=rat(nimi(:,1)); % face sizes as rational numbers 0951 veca=0:1; 0952 plot(0,0,'ok'); 0953 hold on 0954 for j=1:m 0955 i=rotsnub(j); 0956 veca=veca(2)*[1 exp(2i*gamma(i))]; 0957 th=(0:nii(i))*2*pi/nimi(i,1); 0958 sc=[sin(th); 1-cos(th)]; 0959 scv=sc(:,[2 end-1]); % first and last vertices 0960 veci=veca/scv; 0961 plot(real(veci*sc), imag(veci*sc),'-k',real(veci*scv), imag(veci*scv),':b'); 0962 end 0963 hold off 0964 else 0965 if (all(wyd==1) && ~any(md=='w')) || any(md=='f') 0966 0967 % now render the polyhedron 0968 0969 for i=1:f 0970 % patch('Faces',fa{i},'Vertices',vlist,'FaceVertexCData',rgb(i,:),'FaceColor','flat') 0971 fa=zeros(flist(i,5),1); 0972 ix=flist(i,6); 0973 for j=1:numel(fa) 0974 fa(j)=edgeq(ix,1+edgeq(ix,10)); % first vertex of this edge 0975 ix=edgeq(ix,6); % next edge around the face 0976 end 0977 patch(vlist(fa,1),vlist(fa,2),vlist(fa,3),1-((flist(i,1:3)+1)/2).^2,'facealpha',0.95); 0978 end 0979 else 0980 edges=edgeq(edgeq(:,1)<edgeq(:,2),1:2); 0981 plot3(reshape(vlist(edges,1),size(edges))',reshape(vlist(edges,2),size(edges))',reshape(vlist(edges,3),size(edges))','k'); 0982 end 0983 if any(md=='v') 0984 for i=1:v 0985 text(vlist(i,1),vlist(i,2),vlist(i,3),sprintf('%d',i)); 0986 end 0987 end 0988 end 0989 axis equal 0990 if dual 0991 tdual='Dual '; 0992 else 0993 tdual=''; 0994 end 0995 if wyidx==round(wyidx) && wyidx>=1 && wyidx<=size(wys,1) 0996 title(sprintf('%s%s: %s (%s)',tdual,info.wythoff,names{wyidx,1:2})); 0997 else 0998 title(sprintf('%s%s',tdual,info.wythoff)); 0999 end 1000 end