gusucode.com > 《MATLAB图像与视频处理实用案例详解》代码 > 《MATLAB图像与视频处理实用案例详解》代码/第 19 章 基于语音识别的信号灯图像模拟控制技术/voicebox/upolyhedron.m
function [vlist,edgeq,flist,info]=upolyhedron(w,md)
% 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 polyhedron
% This software is based closely on a Mathmatica program described in [1] which, in turn, was based on a
% C program described in [2].
%
% Wythoff Symbol
% p,q,r define a spherical triangle whose angles at the corners are pi/p, pi/q and pi/r;
% this triangle tiles the sphere if repeatedly reflected in its sides. The
% polyhedron vertices are at the reflections of a seed vertex as follows
% where the Vertex configuration gives the polygon orders of the faces around each vertex:
% |p q r : Vertex at a point such that when rotated around any of p,q,r by twice the angle at that
% corner is displaced by the same distance for each corner. This is a snub polyhedron and
% only even numbers of reflections are used to generate vertices. Configuration {3 p 3 q 3 r}
% p|q r : Vertex at p. Configuration = {q r q r ... q r} with 2n terms where n is the numerator of p
% p q|r : Vertex on pq and on the bisector of angle r. Configuration {p 2r q 2r}
% p q r| : Vertex at incentre: meeting point of all the angle bisectors. Configuration {2p 2q 2r}
% |3/2 5/3 3 5/2 This special case is the great dirhombicosidodecahedron. It is a bit weird because
% many of the edges are shared by four faces instead of the usual two.
% If two of p,q,r = 2 then the third is arbitrary (prisms and antiprisms), otherwise only the numerators
% 1:5 can occur, and 4 and 5 cannot occur together. If all are integers then the poyhedron is convex.
%
% References:
% [1] R. E. Maeder. Uniform polyhedra. The Mathematica Journal, 3 (4): 48?7, 1993.
% [2] Z. Har扙l. Uniform solution for uniform polyhedra. Geometriae Dedicata, 47: 57?10, 1993.
% [3] H. S. M. Coxeter, M. S. Longuet-Higgins, and J. C. P. Miller. Uniform polyhedra.
% Philosophical Transactions of the Royal Society A, 246 (916): 401?50, May 1954.
% [4] P. W. Messer. Closed-form expressions for uniform polyhedra and their duals.
% Discrete and Computational Geometry, 27 (3): 353?75, Jan. 2002.
%%%% BUGS and SUGGESTIONS %%%%%%
% (1) we should ensure the "first" edges of the vertices and faces are consistent
% (2) need to sort faces and vertices into a type order
% (3) should ensure that for a non-chiral polyhedron, the vertex polarity alternates
% (4) w=75 does not work
% (5) dual of henispherical poyhedron
% (6) flist not calculated correctly for duals
% (7) add additional stuff into info.*
% (8) vertex figures seem to include additional (duplicated) lines
% (9) sort out when reflected vertices are really rotationally congruent
% (10) could optionally colour by face type
% (11) order alphabetically by noun
% (12) allow abbreviated names + prisms with preceding decimal number
% (13) calculate correct names
% (14) include names for duals
% (15) make "names" etc persistent
% Example slide show:
% for i=1:74, disp(num2str(i)); upolyhedron(i); pause(2); end
% for i=5:10, disp(num2str(i)); for j=1:5, upolyhedron(i+j/10); pause(2); end, end
% Copyright (C) Mike Brookes 1997
% Version: $Id: upolyhedron.m,v 1.4 2010/04/16 07:44:01 dmb Exp $
%
% VOICEBOX is a MATLAB toolbox for speech processing.
% Home page: http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/voicebox.html
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% This program is free software; you can redistribute it and/or modify
% it under the terms of the GNU General Public License as published by
% the Free Software Foundation; either version 2 of the License, or
% (at your option) any later version.
%
% This program is distributed in the hope that it will be useful,
% but WITHOUT ANY WARRANTY; without even the implied warranty of
% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
% GNU General Public License for more details.
%
% You can obtain a copy of the GNU General Public License from
% http://www.gnu.org/copyleft/gpl.html or by writing to
% Free Software Foundation, Inc.,675 Mass Ave, Cambridge, MA 02139, USA.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% [line nn] referes to the kaleido Mathematica program on which this is based
% see http://www.mathconsult.ch/showroom/unipoly/unipoly.html#Images
% Variables used
% adjacent adjacency matrix
% chi characteristic
% cosa cos of the angle subtended by a half edge
% d density
% e number of edges
% even number of even faces to remove
% f number of faces
% fi(n) number of faces of type i
% g order of group
% gamma(n) included spherical triangle angle between centre of face,
% vertex and edge for face type i
% hemiQ =1 for hemispherical faces (includes polyhedron centre)
% incidence faces incident at vertex
% k type of group (2=dihedral, 3=tetrahedral, 4=octahedral, 5=icosahedral)
% m number of edges meeting at each vertex (valence)
% mi(n) Number of faces of type i
% n number of face types
% ni(n) number of edges on i'th face type
% p 1st Wythoff number
% q 2nd Wythoff number
% r 3rd Wythoff number
% rot(m) vertex configuration: anti-clockwise list of face types
% around a vertex
% snub(m) =1 for snub triangles
% snubQ =1 for snub polyhedron
% v number of vertices
% vcoord vertex coordinates
% wy(4) Wythoff Symbol
% wyd(4) Wythoff Symbol Denominators
% wyn(4) Wythoff Symbol Numerators
% names: [name abbreviation synonym dual dual-synonym]
persistent names wys prefs
if ~numel(names)
prefs={'dual '; 'dextro '; 'laevo '};
names={
'Tetrahedron' 'Tet' '' '' ''; % 1
'truncated tetrahedron' 'Tut' '' '' ''; % 2
'octahemioctahedron' 'Oho' '' '' ''; % 3
'tetrahemihexahedron' 'Thah' '' '' ''; % 4
'Octahedron' 'Oct' '' 'Cube' ''; % 5
'cube' 'Cube' '' 'Octahedron' ''; % 6
'cuboctahedron' 'Co' '' 'rhombic dodecahedron' ''; % 7
'truncated octahedron' 'Toe' '' 'tetrakishexahedron' ''; % 8
'truncated cube' 'Tic' '' 'triakisoctahedron' ''; % 9
'rhombicuboctahedron' 'Sirco' '' 'deltoidal icositetrahedron' ''; % 10
'truncated cuboctahedron' '' '' 'disdyakisdodecahedron' ''; % 11
'snub cube' 'Snic' '' 'pentagonal icositetrahedron' ''; % 12
'small cubicuboctahedron' 'Socco' '' 'small hexacronic icositetrahedron' ''; % 13
'great cubicuboctahedron' 'Gocco' '' 'great hexacronic icositetrahedron' ''; % 14
'cubohemioctahedron' 'Cho' '' 'hexahemioctacron' ''; % 15
'cubitruncated cuboctahedron' 'Cotco' '' 'tetradyakishexahedron' ''; % 16
'great rhombicuboctahedron' 'Girco' '' 'great deltoidal icositetrahedron' ''; % 17 or Querco
'small rhombihexahedron' 'Sroh' '' 'small rhombihexacron' ''; % 18
'stellated truncated hexahedron' 'Quith' '' 'great triakisoctahedron' ''; % 19
'great truncated cuboctahedron' 'Quitco' '' 'great disdyakisdodecahedron' ''; % 20
'great rhombihexahedron' 'Groh' '' 'great rhombihexacron' ''; % 21
'icosahedron' 'Ike' '' 'dodecahedron' ''; % 22
'dodecahedron' 'Doe' '' 'icosahedron' ''; % 23
'icosidodecahedron' 'Id' '' 'rhombic triacontahedron' ''; % 24
'truncated icosahedron' 'Ti' '' 'pentakisdodecahedron' ''; % 25
'truncated dodecahedron' 'Tid' '' 'triakisicosahedron' ''; % 26
'rhombicosidodecahedron' 'Srid' '' 'deltoidal hexecontahedron' ''; % 27
'truncated icosidodechedon' 'Grid' 'great rhombiicosidodecahedron' 'disdyakistriacontahedron' ''; % 28
'snub dodecahedron' 'Snid' '' 'pentagonal hexecontahedron' ''; % 29
'small ditrigonal icosidodecahedron' 'Sidtid' '' 'small triambic icosahedron' ''; % 30
'small icosicosidodecahedron' 'Siid' '' 'small icosacronic hexecontahedron' ''; % 31
'small snub icosicosidodecahedron' 'Seside' '' 'small hexagonal hexecontahedron' ''; % 32
'small dodecicosidodecahedron' 'Saddid' '' 'small dodecacronic hexecontahedron' ''; % 33
'small stellated dodecahedron' 'Sissid' '' 'great dodecahedron' ''; % 34
'great dodecahedron' 'Gad' '' 'small stellated dodecahedron' ''; % 35
'dodecadodecahedron' 'Did' '' 'medial rhombic triacontahedron' ''; % 36
'truncated great dodecahedron' 'Tigid' '' 'small stellapentakisdodecahedron' ''; % 37
'rhombidodecadodecahedron' 'Raded' '' 'medial deltoidal hexecontahedron' ''; % 38
'small rhombidodecahedron' 'Sird' '' 'small rhombidodecacron' ''; % 39
'snub dodecadodecahedron' 'Siddid' '' 'medial pentagonal hexecontahedron' ''; % 40
'ditrigonal dodecadodecahedron' 'Ditdid' '' 'medial triambic icosahedron' ''; % 41
'great ditrigonal dodecicosidodecahedron' 'Gidditdid' '' 'great ditrigonal dodecacronic hexecontahedron' ''; % 42
'small ditrigonal dodecicosidodecahedron' 'Sidditdid' '' 'small ditrigonal dodecacronic hexecontahedron' ''; % 43
'icosidodecadodecahedron' 'Ided' '' 'medial icosacronic hexecontahedron' ''; % 44
'icositruncated dodecadodecahedron' 'Idtid' '' 'tridyakisicosahedron' ''; % 45
'snub icosidodecadodecahedron' 'Sided' '' 'medial hexagonal hexecontahedron' ''; % 46
'great ditrigonal icosidodecahedron' 'Gidtid' '' 'great triambic icosahedron' ''; % 47
'great icosicosidodecahedron' 'Giid' '' 'great icosacronic hexecontahedron' ''; % 48
'small icosihemidodecahedron' 'Seihid' '' 'small icosihemidodecacron' ''; % 49
'small dodecicosahedron' 'Siddy' '' 'small dodecicosacron' ''; % 50
'small dodecahemidodecahedron' 'Sidhid' '' 'small dodecahemidodecacron' ''; % 51
'great stellated dodecahedron' 'Gissid' '' 'great icosahedron' ''; % 52
'great icosahedron' 'Gike' '' 'great stellated dodecahedron' ''; % 53
'great icosidodecahedron' 'Gid' '' 'great rhombic triacontahedron' ''; % 54
'great truncated icosahedron' 'Tiggy' '' 'great stellapentakisdodecahedron' ''; % 55
'rhombicosahedron' 'Ri' '' 'rhombicosacron' ''; % 56
'great snub icosidodecahedron' 'Gosid' '' 'great pentagonal hexecontahedron' ''; % 57
'small stellated truncated dodecahedron' 'Quitsissid' '' 'great pentakisdodekahedron' ''; % 58
'truncated dodecadodecahedron' 'Quitdid' '' 'medial disdyakistriacontahedron' ''; % 59
'inverted snub dodecadodecahedron' 'Isdid' '' 'medial inverted pentagonal hexecontahedron' ''; % 60
'great dodecicosidodecahedron' 'Gaddid' '' 'great dodecacronic hexecontahedron' ''; % 61
'small dodecahemicosahedron' 'Sidhei' '' 'small dodecahemicosacron' ''; % 62
'great dodecicosahedron' 'Giddy' '' 'great dodecicosacron' ''; % 63
'great snub dodecicosidodecahedron' 'Gisdid' '' 'great hexagonal hexecontahedron' ''; % 64
'great dodecahemicosahedron' 'Gidhei' '' 'great dodecahemicosacron' ''; % 65
'great stellated truncated dodecahedron' 'Quitgissid' '' 'great triakisicosahedron' ''; % 66
'great rhombicosidodecahedron' 'Qrid' '' 'great deltoidal hexecontahedron' ''; % 67 or Qrid
'great truncated icosidodecahedron' 'Gaquatid' '' 'great disdyakistriacontahedron' ''; % 68
'great inverted snub icosidodecahedron' 'Gisid' '' 'great inverted pentagonal hexecontahedron' ''; % 69
'great dodecahemidodecahedron' 'Gidhid' '' 'great dodecahemidodecacron' ''; % 70
'great icosihemidodecahedron' 'Geihid' '' 'great icosihemidodecacron' ''; % 71
'small retrosnub icosicosidodecahedron' 'Sirsid' '' 'small hexagrammic hexecontahedron' ''; % 72
'great rhombidodecahedron' 'Gird' '' 'great rhombidodecacron' ''; % 73
'great retrosnub icosidodecahedron' 'Girsid' '' 'great pentagrammic hexecontahedron' ''; % 74
'great dirhombicosidodecahedron' 'Gidrid' '' 'great dirhombicosidodecacron' ''; % 75
'pentagonal prism' 'Pip' '' '' ''; % 76
'pentagonal antiprism' 'Pap' '' '' ''; % 77
'pentagrammic prism' 'Stip' '' '' ''; % 78
'pentagrammic antiprism' 'Stap' '' '' ''; % 79
'pentagrammic crossed antiprism' 'Starp' '' '' ''; % 80
'triangular prism' '' '' '' ''; % 81
'triangular antiprism' '' '' '' ''; % 82
'square prism' '' '' '' ''; % 83
'square antiprism' '' '' '' ''}; % 84
% could multiply by 12 to make integer
wys=[
2, 3, 2, 3; % 1 'tetrahedron' 'Tet';
3, 2, 3, 3; % 2 'truncated tetrahedron' 'Tut';
3, 3/2, 3, 3; % 3 'octahemioctahedron' 'Oho';
3, 3/2, 3, 2; % 4 'tetrahemihexahedron' 'Thah';
2, 4, 2, 3; % 5 'octahedron' 'Oct';
2, 3, 2, 4; % 6 'cube' 'Cube';
2, 2, 3, 4; % 7 'cuboctahedron' 'Co';
3, 2, 4, 3; % 8 'truncated octahedron' 'Toe';
3, 2, 3, 4; % 9 'truncated cube' 'Tic';
3, 3, 4, 2; % 10 'rhombicuboctahedron' 'Sirco';
4, 2, 3, 4; % 11 'truncated cuboctahedron' '?';
1, 2, 3, 4; % 12 'snub cube' 'Snic';
3, 3/2, 4, 4; % 13 'small cubicuboctahedron' 'Socco';
3, 3, 4, 4/3; % 14 'great cubicuboctahedron' 'Gocco';
3, 4/3, 4, 3; % 15 'cubohemioctahedron' 'Cho';
4, 4/3, 3, 4; % 16 'cubitruncated cuboctahedron' 'Cotco';
3, 3/2, 4, 2; % 17 'great rhombicuboctahedron' 'Girco?';
4, 3/2, 2, 4; % 18 'small rhombihexahedron' 'Sroh';
3, 2, 3, 4/3; % 19 'stellated truncated hexahedron' 'Quith';
4, 4/3, 2, 3; % 20 'great truncated cuboctahedron' 'Quitco';
4, 4/3, 3/2, 2; % 21 'great rhombihexahedron' 'Groh';
2, 5, 2, 3; % 22 'icosahedron' 'Ike';
2, 3, 2, 5; % 23 'dodecahedron' 'Doe';
2, 2, 3, 5; % 24 'icosidodecahedron' 'Id';
3, 2, 5, 3; % 25 'truncated icosahedron' 'Ti';
3, 2, 3, 5; % 26 'truncated dodecahedron' 'Tid';
3, 3, 5, 2; % 27 'rhombicosidodecahedron' 'Srid';
4, 2, 3, 5; % 28 'truncated icosidodecahedron' 'Grid'; or 'great rhombiicosidodecahedron' Grid
1, 2, 3, 5; % 29 'snub dodecahedron' 'Snid';
2, 3, 5/2, 3; % 30 'small ditrigonal icosidodecahedron' 'Sidtid';
3, 5/2, 3, 3; % 31 'small icosicosidodecahedron' 'Siid';
1, 5/2, 3, 3; % 32 'small snub icosicosidodecahedron' 'Seside';
3, 3/2, 5, 5; % 33 'small dodecicosidodecahedron' 'Saddid';
2, 5, 2, 5/2; % 34 'small stellated dodecahedron' 'Sissid';
2, 5/2, 2, 5; % 35 'great dodecahedron' 'Gad';
2, 2, 5/2, 5; % 36 'dodecadodecahedron' 'Did';
3, 2, 5/2, 5; % 37 'truncated great dodecahedron' 'Tigid';
3, 5/2, 5, 2; % 38 'rhombidodecadodecahedron' 'Raded';
4, 2, 5/2, 5; % 39 'small rhombidodecahedron' 'Sird';
1, 2, 5/2, 5; % 40 'snub dodecadodecahedron' 'Siddid';
2, 3, 5/3, 5; % 41 'ditrigonal dodecadodecahedron' 'Ditdid';
3, 3, 5, 5/3; % 42 'great ditrigonal dodecicosidodecahedron' 'Gidditdid'
3, 5/3, 3, 5; % 43 'small ditrigonal dodecicosidodecahedron' 'Sidditdid'
3, 5/3, 5, 3; % 44 'icosidodecadodecahedron' 'Ided';
4, 5/3, 3, 5; % 45 'icositruncated dodecadodecahedron' 'Idtid';
1, 5/3, 3, 5; % 46 'snub icosidodecadodecahedron' 'Sided';
2, 3/2, 3, 5; % 47 'great ditrigonal icosidodecahedron' 'Gidtid';
3, 3/2, 5, 3; % 48 'great icosicosidodecahedron' 'Giid';
3, 3/2, 3, 5; % 49 'small icosihemidodecahedron' 'Seihid';
4, 3/2, 3, 5; % 50 'small dodecicosahedron' 'Siddy';
3, 5/4, 5, 5; % 51 'small dodecahemidodecahedron' 'Sidhid';
2, 3, 2, 5/2; % 52 'great stellated dodecahedron' 'Gissid';
2, 5/2, 2, 3; % 53 'great icosahedron' 'Gike';
2, 2, 5/2, 3; % 54 'great icosidodecahedron' 'Gid';
3, 2, 5/2, 3; % 55 'great truncated icosahedron' 'Tiggy';
4, 2, 5/2, 3; % 56 'rhombicosahedron' 'Ri';
1, 2, 5/2, 3; % 57 'great snub icosidodecahedron' 'Gosid';
3, 2, 5, 5/3; % 58 'small stellated truncated dodecahedron' 'Quitsissid'
4, 5/3, 2, 5; % 59 'truncated dodecadodecahedron' 'Quitdid';
1, 5/3, 2, 5; % 60 'inverted snub dodecadodecahedron' 'Isdid';
3, 5/2, 3, 5/3; % 61 'great dodecicosidodecahedron' 'Gaddid';
3, 5/3, 5/2, 3; % 62 'small dodecahemicosahedron' 'Sidhei';
4, 5/3, 5/2, 3; % 63 'great dodecicosahedron' 'Giddy';
1, 5/3, 5/2, 3; % 64 'great snub dodecicosidodecahedron' 'Gisdid';
3, 5/4, 5, 3; % 65 'great dodecahemicosahedron' 'Gidhei';
3, 2, 3, 5/3; % 66 'great stellated truncated dodecahedron' 'Quitgissid'
3, 5/3, 3, 2; % 67 'quasirhombicosidodecahedron' 'Qrid'; 'great rhombicosidodecahedron' 'Nonconvex great rhombicosidodecahedron
4, 5/3, 2, 3; % 68 'great truncated icosidodecahedron' 'Gaquatid';
1, 5/3, 2, 3; % 69 'great inverted snub icosidodecahedron' 'Gisid';
3, 5/3, 5/2, 5/3; % 70 'great dodecahemidodecahedron' 'Gidhid';
3, 3/2, 3, 5/3; % 71 'great icosihemidodecahedron' 'Geihid';
1, 3/2, 3/2, 5/2; % 72 'small retrosnub icosicosidodecahedron' 'Sirsid';
4, 3/2, 5/3, 2; % 73 'great rhombidodecahedron' 'Gird';
1, 3/2, 5/3, 2; % 74 'great retrosnub icosidodecahedron' 'Girsid';
5, 3/2, 5/3, 3; % 75 'great dirhombicosidodecahedron' 'Gidrid';
3, 2, 5, 2; % 76 'pentagonal prism' 'Pip';
1, 2, 2, 5; % 77 'pentagonal antiprism' 'Pap';
3, 2, 5/2, 2; % 78 'pentagrammic prism' 'Stip';
1, 2, 2, 5/2; % 79 'pentagrammic antiprism' 'Stap';
1, 2, 2, 5/3; % 80 'pentagrammic crossed antiprism' 'Starp'
3, 2, 3, 2; % 81 'triangular prism' ;
1, 2, 2, 3; % 82 'triangular antiprism' ;
3, 2, 4, 2; % 83 'square prism' ;
1, 2, 2, 4]; % 84 'square antiprism' ;
end
if nargin<2
md='';
end
dual=0; % dual polyhedron
dextro=0; % reflected (for snub polyhedra only)
if ischar(w)
i=0;
while i<length(prefs)
i=i+1;
if length(w)>length(prefs{i}) && strcmpi(w(1:length(prefs{i})),prefs{i})
switch i
case 1
dual=1;
case 2
dextro=1;
end
w=w(length(prefs{i})+1:end);
i=0;
end
end
wyidx=find(strcmpi(w,names(:)),1);
if isempty(wyidx)
% check for prism names here
error('Cannot find %s',w);
end
if wyidx>3*size(names,1)
dual=1-dual;
end
wyidx=1+rem(wyidx-1,size(names,1));
wy=wys(wyidx,:);
else
if length(w)==1
dual=w<0;
wyidx=abs(w);
wyjdx=floor(wyidx);
wykdx=round(10*(wyidx-wyjdx));
switch wykdx
case 1 % prism
wy=[3 2 wyjdx 2];
case 2 % antiprism
wy=[1 2 2 wyjdx];
case 3 % grammic prism
wy=[3 2 wyjdx/2 2];
case 4 % grammic antiprism
wy=[1 2 2 wyjdx/2];
case 5 % grammic crossed antiprism
wy=[1 2 2 wyjdx/3];
otherwise
if wyjdx>=1 && wyjdx<=size(wys,1)
wy=wys(wyjdx,:);
else
error('Polyhedron number out of range');
end
end
elseif length(w)==4
vbar=find(w<=0,1);
if ~numel(vbar)
error('Invalid Wythoff symbol: %g %g %g %g %g',w);
end
dual=mod(w(vbar),2); % least significant bit indicates dual
dextro=mod(1+w(vbar),4)>=2; % second bit indicates reflected version
wy=[vbar w(1:vbar-1) w(vbar+1:4)];
[xx,wyidx]=min(sum((wys-repmat(wy,size(wys,1),1)).^2,2));
if abs(xx)>1e-3
if sum(wy==2)<2
error('Invalid Wythoff symbol: %g %g %g %g %g',w);
else
% need to sort out prism names here
end
end
elseif length(w)==5
if w(1)<=0 && all(round(12*w(2:end))==[18 40 36 30]) % [3/2 5/3 3 5/2]
dual=w(1)<0;
wyidx=75;
wy=wys(wyidx,:);
else
error('Invalid Wythoff symbol: %g %g %g %g %g',w);
end
else
error('Invalid polyhedron specification');
end
end
[wyn,wyd]=rat(wy(2:4)); % convert to rational numbers
wy(2:4)=wyn./wyd; % force exact rational values
p=wy(2);
q=wy(3);
r=wy(4);
hemiQ = 0; % includes hemispherical faces that go through polyhedron centre [ p q | r] with pq=p+q
onesidedQ = 0; % one-sided polyhedron (aka: non-orientable) [ p q r |] with exactly one of p,q,r having an even denominator
evenQ = 0; % identifies which of p, q, r has an even denominator
even=1;
snubQ = 0; % snub polyhedron: [ | p q r ]
allrot = 1; % all vertices a congruent with rotations (no reflections needed)
% call to AnalyseWythoff[line 105]
k= max(wyn);
if sum(wy(2:4)==2)>=2 % check if it is a prism
% this is a prism - treat specially [note mathematica line 86 has redundant chack for 2 >5]
g=4*k; % order of group
k=2;
else % not a prism: only numerators 1,2,3,4,5 are allowed
if (any(wy(2:4)<=1) || (k>5) || any(wyn==4) && any(wyn==5))
error('Invalid Wythoff numbers [%d/%d %d/%d %d/%d]',[wyn;wyd]);
end
g=24*k/(6-k); % order of group
end
if abs(wy(1))==5 % special case
chi = 0;
d = 0;
else
chi = (sum(wyn.^(-1))-1)*g/2;
d = (sum(wy(2:4).^(-1))-1)*g/4;
if d<0
error('density < 0');
end
end
if abs(wy(1))==5
error('great dirhombicosidodecahedron not implemented');
end
switch abs(wy(1))
case {1,5} % [ | p q r ] snub polyhedron
n=4; % number of face types
m=6; % valence of vertices
v=g/2; % number of vertices
% ni=[3 wy(2:4)]; % type of each face
% mi=[3 1 1 1]; % number of faces of each type
nimi=[3 wy(2:4); 3 1 1 1]';
% rot = [1 2+dextro 1 3-dextro 1 4];
snubQ=1;
% snub=[1 0 1 0 1 0];
rotsnub=[1 2+dextro 1 3-dextro 1 4; repmat([1 0],1,3)]';
case 2 % [ p | q r ]
n=2;
m=2*wyn(1);
v=g/m;
% ni=wy(3:4);
% mi=[p p]; % these face counts may be fractional: m counts their numerators
% we could ill in the numerator denominator columns, nimi(:,3:4) here instead of later
nimi=[wy(3:4); p p]';
% rot = repmat(1:2,1,wyn(1));
rotsnub = repmat([1 0; 2 0],wyn(1),1);
case 3 % [ p q | r ]
n=3;
m=4;
v=g/2;
% ni=[2*r wy(2:3)];
% mi = [2 1 1];
nimi=[2*r wy(2:3); 2 1 1]';
% rot = [1 2 1 3];
rotsnub = [1 2 1 3; zeros(1,4)]';
if abs(p-q/(q-1))<1e-6
hemiQ=1;
d=0;
if (p~=2 && ~(wy(4)==3 && any(wy(2:3)==3)))
onesidedQ=1;
v=v/2;
chi=chi/2;
end
end
case 4 % [ p q r | ]
n=3;
m=3;
v=g;
% ni=2*wy(2:4);
% mi=ones(1,3);
nimi=[2*wy(2:4); ones(1,3)]';
% rot=1:3;
rotsnub=[1:3; zeros(1,3)]';
allrot=(wy(2)==wy(3)) || (wy(2)==wy(4)) || (wy(3)==wy(4)); % isosceles so all
evenden=find(~rem(wyd,2)); % check for even denominators
if (length(evenden)>1)
error('Multiple even denominators are not allowed');
end
if ~isempty(evenden)
even = evenden;
evenQ=1;
onesidedQ = 1;
d=0;
v=v/2;
chi = chi - g/wyn(even)/2; % check this *****
end
otherwise
error('Invalid polyhedron type %d',wy(1));
end
% [line 155] call sortAndMerge [sortAndMerge: line 220]
% save stuff to prevent overwriting
n_1=n;
m_1=m;
% now do sort merge based on rotsnub and nimi
% rotsnub(:,2) = [rot snub]
% nimi(:,4) = [ni mi mi-num mi-den]
nimi(nimi(:,1)==2,1)=-2; % make equal to -2 to force to bottom of list
[nimi,inim]=sortrows(nimi,-1);
jnim=zeros(n,1);
jnim(inim)=1:n;
msk=[~0; nimi(2:end,1)<nimi(1:end-1,1)]; % identify distinct values
cmsk=cumsum(msk); % destination of each value
nimi(msk,2)=sparse(1,cmsk,nimi(:,2)); % add up repreated counts
nimi=nimi(msk,:); % select just the distinct values
% cmsk(jnim) maps original to new positions
[nimi(:,3),nimi(:,4)]=rat(nimi(:,2));
rotsnub(:,1)=cmsk(jnim(rotsnub(:,1)));
even=cmsk(jnim(even));
if nimi(end,1)<0 % remove digons
if size(nimi,1)==1
error('Degenerate polyhedron (digons only)');
end
m=m-nimi(end,3); % note that m=sum(nimi(:,3)) not sum(nimi(:,2)) as you might expect
nimi(end,:)=[];
n=size(nimi,1);
rotsnub(rotsnub(:,1)>n,:)=[]; % abolish references to digons
end
% original sort and merge
% [ss,tr]=sort(-ni); % sort into descending order
% itr=zeros(1,n);
% itr(tr)=1:n;
% ni=ni(tr);
% mi=mi(tr);
% rot = itr(rot);
% if evenQ
% even = itr(even);
% end
% % now merge equal faces
% i=1;
% while i<n
% if ni(i)==ni(i+1)
% mi(i)=mi(i)+mi(i+1);
% mi(i+1)=[];
% ni(i+1)=[];
% n=n-1;
% even=even-(even>i);
% rot = rot - (rot>i);
% else
% i=i+1;
% end
% end
% [mii,mij]=rat(mi); % find numerator
% i=find(ni==2); % find digons
% if ~isempty(i)
% if n==1
% error('Degenerate (digons only)');
% end
% i=i(1);
% m=m-mii(i); % reduce total valance
% ni(i)=[];
% mi(i)=[];
% mii(i)=[];
% even=even-(even>i);
% if snubQ
% snub(rot==i)=[];
% end
% rot(rot==i)=[];
% rot = rot - (rot>i);
% n=n-1;
% end
%
% % check that new version is correct
%
%
% if any(nimi(:,1:3)~=[ni' mi' mii'])
% error('nimi has errors');
% end
% if snubQ
% if any(rotsnub~=[rot' snub'])
% error('rotsnub has errors');
% end
% else
% if any(rotsnub(:,1)~=rot')
% error('rotsnub has errors');
% end
% end
% if evenQ && even~=even_1
% error('even has errors');
% end
% [line 159] Solve fundamental equations [line 266]
% doesn't converge well for a dodecahedron
% we could avoid the iteration when n=1
%
% We want to solve the following set of equations for cosa and gammai:
% (a) sum(mi.*gammai)=pi
% (b) cos(alphai) = cosa*sin(gammai)
% where ni, mi, alphai and gammai are vectors of length m, the number of edges at a vertex
% ni gives the number of sides in each face type in decreasing order (possibly fractional)
% mi is the repretition count of each face type (possibly fractional)
% alphai=pi * ni.^(-1) is the angle subtended by a half edge at the centre of a face
% gammai (> pi/2 - alphai) is half the spherical angle subtended by a face at the vertex
% cosa (< 1) is the distance of the edge centre from the origin if the radius of the vertex is unity
% we solve this iteratively using gammai(1) as the independent variable
alphai = pi*nimi(:,1).^(-1); % half of external angle of polygon
calphai = cos(alphai);
ca1 = calphai(1);
calphai(1)=[];
% initial values
gammai = pi/2 - alphai; % half of internal angle of polygon
delta = pi - nimi(:,2)'*gammai;
% we could initialize it to be exact for a single face type
% g1n=gammai(1)*pi/(mi*gammai')
% cosa = ca1/sin(g1n);
% gammai = [g1n asin(calphai/cosa)];
% delta = pi - mi*gammai';
% should check here to see if delta is better than before and g1n is valid
iter=0;
while (abs(delta) > 1e-10)
g1n = gammai(1) + delta*tan(gammai(1))/(nimi(:,2)'*tan(gammai));
if (g1n<0 || g1n>pi)
error ('Gamma out of range');
end
cosa = ca1/sin(g1n);
gammai = [g1n; asin(calphai/cosa)];
delta = pi - nimi(:,2)'*gammai;
iter=iter+1;
end
gamma = gammai;
cosa = ca1/sin(gammai(1));
% [line 165] postprocess special cases
if evenQ
nimi(even,:)=[];
gamma(even)=[];
nh=n-1;
n=2*nh;
m=4;
% ni=[ni 1+fliplr(ni-1).^(-1)];
nimi=repmat(nimi,2,1);
nimi(nh+1:end,1)=1+(nimi(nh:-1:1,1)-1).^(-1);
gamma=[gamma; -flipud(gamma)];
% mi=repmat(3-nh,1,n);
% mii=mi;
nimi(:,2:3)=repmat(3-nh,n,2);
rotsnub=[1 nh n 1+nh; zeros(1,4)]';
end
if wy(1)==5
error('not yet implemented');
% needs to use rotsnub and nimi
n=5;
m=8;
hemiQ = 1;
d=0;
mi=[4 1 1 1 1];
mii=mi;
ni=[4 ni(1:3) ni(1)/(ni(1)-1)];
gamma=[pi/2 gamma(1:3) -gamma(1)];
rot=[rot+[0 1 0 1 0 1] 1 5]; % [1 4 1 3 1 2 1 5]
snub=[snub 1 0];
end
% [line 197] count vertices and faces [count: line 286]
e = m*v/2; % number of edges = valency * vertices /2
[nii,nij]=rat(nimi(:,1)); % find numerators
fi = v*nimi(:,3)./nii;
f = sum(fi);
if (d>0) && (gamma(1)> pi/2) % [might be better to use cycles rather than radians for gamma]
d = fi(1)-d;
end
if wy(1)==5
chi = v - e + f; % Euler characteristic
end
% [line 199] generate vertex coordinates and adjacency matrix [vertices: line 297]
% we currently scale it so the edge mid-points have unit radius
adj=zeros(v,m); % j'th vertex adjacent to vertex i anti-clockwise;
frot = zeros(1,v);
rev = zeros(v,1);
vlist = zeros(v+1,7); % vertex information: [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)
vlist(:,5)=m; % all vertices have the same valency
v1 = [0 0 1]/cosa; % first vertex
frot(1) = 1;
adj(1,1) = 2;
v2 = [2*cosa*sqrt(1-cosa^2), 0, 2*cosa^2-1]/cosa; % second vertex
if snubQ
frot(2) = m*(1-rotsnub(m,2))+rotsnub(m,2); % inefficient
adj(2,1)=1;
else % reflexible
frot(2) = 1;
rev(2) = 1;
adj(2,m)=1;
end
vlist(1,1:3)=v1;
vlist(2,1:3)=v2;
nv = 2;
i = 1;
skew=zeros(3,3);
skewp=[6 7 2]; % index of positive entries
skewn=[8 3 4]; % index of negative entries
cosg=cos(2*gamma);
sing=sin(2*gamma);
eye3=eye(3);
% veqth=cos(acos(cosa)/(d+1)); % threshold for vertex equality test = cosa * |vlist(1,1:3)|^2
veqth=0.999999/(cosa^2); % threshold for vertex equality test = cosa * |vlist(1,1:3)|^2
while i<=nv % loop for each vertex
if rev(i)
one = -1;
start = m-1;
limit = 1;
else
one = 1;
start = 2;
limit = m;
end
k = frot(i); % rotation to use first
v1 = vlist(i,1:3); % the centre of rotation
v1 = v1/sqrt(v1*v1'); % normalize to unit length [not clear why we don't make unit length in the first place]
v2 = vlist(adj(i,start-one),1:3);
for j=start:one:limit
% R=wwT+cos(x)(I-wwT)+sin(x)SKEW(w)
% SKEW(a) = [0 -a3 a2; a3 0 -a1; -a2 a1 0]
skew(skewp)=v1;
skew(skewn)=-v1;
rsym=v1'*v1;
rotk=rotsnub(k,1);
rotmat=rsym+cosg(rotk)*(eye3-rsym)-one*sing(rotk)*skew;
v2=v2*rotmat; % rotate v2 poition around v1
vlist(nv+1,1:3)=v2; % add into list in case it is good
nvi=find(vlist(:,1:3)*v2'>veqth,1); % take the first matching vertex
% cmatch=1-vlist(nvi,:)*v2'*cosa^2 % diagnostic printout for vertex match test (0 for perfect match)
adj(i,j)=nvi; % save as next vertex
lastk = k;
k=1+mod(k,m); % increment k circularly in the range 1:m
if nvi>nv % we have a new vertex
if snubQ % if a snub polyhedron
% Mathematica: frot[[nvi]] = If[ !sn[[lastk]], lastk, If[ !sn[[k]], next[k, m], k ] ];
% A snub triangle plays the role of the next available snub triangle for the next vertex
frot(nvi)=lastk+rotsnub(lastk,2)*(rotsnub(k,2)*k-lastk+(1-rotsnub(k,2))*(1+mod(k,m)));
rev(nvi)=0; % snub polyhedra always have rev=0
adj(nvi,1)=i;
else
frot(nvi)=k;
rev(nvi)=(1+one)/2; % = 1-rev(i)
adj(nvi,1+(m-1)*rev(nvi))=i;
end
nv=nvi;
if nv>v
error('Too many vertices found');
end
end
end
i=i+1;
end
if (nv~=v)
error('Not all vertices found');
end
vlist(v+1,:)=[]; % remove the dummy extra vertex
vlist(:,7)=1-2*rev; % vertex types all all +1 or -1
% construct edge map
% edgeq: 1=v1 2=v2 3=f1 4=f2 5=ev1 6=ef1 7=er 8=z 9=sf 10=sv]
% 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
edgeq=zeros(v*m,10); % OLD *** ===[reverse_edge start_vertex left_face next_edge_at_vertex next_edge_on_face]
edgeq(:,1:2)=[repmat((1:v)',m,1) adj(:)]; % 2=v_start 3=v_end
edgeq(:,3:4)=edgeq(:,1:2)*[v 1; 1 v]; % encode start and end as a single integer
[xx,ia]=sort(edgeq(:,3));
[xx,ib]=sort(edgeq(:,4));
edgeq(:,3:4)=0;
ib(ib)=1:v*m; % make reverse index
edgeq(:,7)=ia(ib); % index to reverse direction edge: 1=e_reverse
edgeq(:,5)=[v+1:v*m 1:v]; % adjacent edge anti-clockwise around vertex: 6=e_next_at_start
ia(edgeq(:,5))=1:v*m; % adjacent edge clockwise around vertex; alternatively ia = mod((1:v*m)-v,v*m)'
edgeq(:,6)=ia(edgeq(:,7)); % adjacent edge anti-clockwise around face [but not necessarily correct
fpt=zeros(f,2); % edge count and pointer to an edge on a face
iff=0; % face index
if onesidedQ
edgeq(:,8)=reshape(mod(repmat(frot',1,m)+repmat((1:m),v,1),2),v*m,1); %face type modulo 2
edgeq(:,8)=abs(edgeq(:,8)-edgeq(edgeq(:,6),8)); % 1 = twisted edge
while iff<f
iee=find(edgeq(:,3)==0,1); % find an edge without a left face
if ~numel(iee)
error('Not enough faces found');
end
iff=iff+1;
fpt(iff,2)=iee;
nee=0;
flip=abs((edgeq(iee,6)==edgeq(edgeq(iee,7),5))-edgeq(iee,8)); % check if this initial edge is flipped on output
while edgeq(iee,3)~=iff
if edgeq(iee,3) % use the reverse edge if this one is already taken
iee=edgeq(iee,7);
edgeq(pee,6)=iee; % correct the previous exit edge taken
edgeq(iee,10)=1; % use it in the reverse direction
edgeq(iee,6)=ia(iee); % normal exit for this direction
jee=edgeq(iee,5); % alternate exit edge if flipping
else
if flip && ~edgeq(iee,4) % we are entering on the usual reverse exit
edgeq(edgeq(iee,7),6)=edgeq(iee,5); % So give the reverse exit a valid path
end
jee=edgeq(edgeq(iee,7),5); % alternate exit edge if flipping
end
if edgeq(iee,3)
error('Edge already in use');
end
edgeq(iee,3)=iff;
edgeq(edgeq(iee,7),4)=iff; % mark right face in reverse edge
nee=nee+1;
flip=abs(flip-edgeq(iee,8));
if flip % use alternate edge
edgeq(iee,6)=jee; % mark actual edge used
end
pee=iee; % save previous iee value
iee=edgeq(iee,6); % step around the face
end
fpt(iff,1)=nee;
end
else % a normal two-sided polygon
while iff<f
iee=find(edgeq(:,3)==0,1); % find an edge without a left face
iff=iff+1;
fpt(iff,2)=iee;
nee=0;
while edgeq(iee,3)~=iff
if edgeq(iee,3)
error('Edge already in use');
end
edgeq(iee,3)=iff;
edgeq(edgeq(iee,7),4)=iff; % mark right face in reverse edge
nee=nee+1;
iee=edgeq(iee,6); % step around the face
end
fpt(iff,1)=nee;
end
end
% [(1:v*m)' edgeq]
vlist(:,4)=sqrt(sum(vlist(:,1:3).^2,2)); % fill in the radii
vlist(:,6)=(1:v)';
% now fill in the flist array
% [x y z d n e t] for each face
flist=zeros(f,7);
[ia,ib]=sort(edgeq(:,3)); % finst all the edges belonging to each face
ix=[1; 1+find(ia(2:end)>ia(1:end-1)); 2*e+1];
flist(:,6)=ib(ix(1:f)); % pointer to first edge for each face
flist(:,5)=ix(2:end)-ix(1:end-1); % size of each face
% edgeq: 1=v1 2=v2 3=f1 4=f2 5=ev1 6=ef1 7=er 8=z 9=sf 10=sv]
tedge=flist(:,6); % this edge
nedge=edgeq(tedge,6); % next edge
vmid=vlist(edgeq(tedge+2*e*(1-edgeq(tedge,10))),1:3);
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);
flist(:,1:3)=flist(:,1:3)./repmat(sqrt(sum(flist(:,1:3).^2,2)),1,3);
flist(:,4)=sum(flist(:,1:3).*vmid,2);
% now check for dual
if dual
vlisto=vlist; % save vlist
vlist=flist; % old faces are new vertices
flist=vlisto;
% edgeq: 1=v1 2=v2 3=f1 4=f2 5=ev1 6=ef1 7=er 8=z 9=sf 10=sv]
edgeq=edgeq(:,[3 4 2 1 6 5 7 8 10 9]); % swap vertices and faces in the edge list
erev=edgeq(:,7); % reverse edge
edgeq(:,6)=erev(edgeq(erev,6));
flist(:,6)=erev(flist(:,6));
if hemiQ % cannot invert through centre
error('Cannot take dual');
else
vlist(:,4)=vlist(:,4).^(-1);
end
vlist(:,1:3)=vlist(:,1:3).*repmat(vlist(:,4),1,3);
v=size(vlist,1);
vlist(:,1:3)=vlist(:,1:3)-repmat(mean(vlist(:,1:3),1),v,1);
f=size(flist,1); % recalculate face positions from scratch
tedge=flist(:,6); % this edge
nedge=edgeq(tedge,6); % next edge
vmid=vlist(edgeq(tedge+2*e*(1-edgeq(tedge,10))),1:3);
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);
flist(:,1:3)=flist(:,1:3)./repmat(sqrt(sum(flist(:,1:3).^2,2)),1,3);
flist(:,4)=sum(flist(:,1:3).*vmid,2);
end
info.snub=snubQ>0;
info.onesided=onesidedQ>0;
info.hemi=hemiQ>0;
% create Wythoff string
wystr='';
j=0;
for i=1:4
if i==abs(wy(1))
wystr=[wystr '| '];
j=1;
else
if (wyd(i-j)==1)
wystr=[wystr num2str(wyn(i-j)) ' '];
else
wystr=[wystr num2str(wyn(i-j)) '/' num2str(wyd(i-j)) ' '];
end
end
end
info.wythoff=wystr(1:end-1);
info.vef=[v e f];
info.chi=chi; % Euler Characteristic
% now draw an image if requested
if ~nargout || any(md=='g')
clf;
if any(md=='t')
[nii,nij]=rat(nimi(:,1)); % face sizes as rational numbers
veca=0:1;
plot(0,0,'ok');
hold on
for j=1:m
i=rotsnub(j);
veca=veca(2)*[1 exp(2i*gamma(i))];
th=(0:nii(i))*2*pi/nimi(i,1);
sc=[sin(th); 1-cos(th)];
scv=sc(:,[2 end-1]); % first and last vertices
veci=veca/scv;
plot(real(veci*sc), imag(veci*sc),'-k',real(veci*scv), imag(veci*scv),':b');
end
hold off
else
if (all(wyd==1) && ~any(md=='w')) || any(md=='f')
% now render the polyhedron
for i=1:f
% patch('Faces',fa{i},'Vertices',vlist,'FaceVertexCData',rgb(i,:),'FaceColor','flat')
fa=zeros(flist(i,5),1);
ix=flist(i,6);
for j=1:numel(fa)
fa(j)=edgeq(ix,1+edgeq(ix,10)); % first vertex of this edge
ix=edgeq(ix,6); % next edge around the face
end
patch(vlist(fa,1),vlist(fa,2),vlist(fa,3),1-((flist(i,1:3)+1)/2).^2,'facealpha',0.95);
end
else
edges=edgeq(edgeq(:,1)<edgeq(:,2),1:2);
plot3(reshape(vlist(edges,1),size(edges))',reshape(vlist(edges,2),size(edges))',reshape(vlist(edges,3),size(edges))','k');
end
if any(md=='v')
for i=1:v
text(vlist(i,1),vlist(i,2),vlist(i,3),sprintf('%d',i));
end
end
end
axis equal
if dual
tdual='Dual ';
else
tdual='';
end
if wyidx==round(wyidx) && wyidx>=1 && wyidx<=size(wys,1)
title(sprintf('%s%s: %s (%s)',tdual,info.wythoff,names{wyidx,1:2}));
else
title(sprintf('%s%s',tdual,info.wythoff));
end
end