1discgrp(5gv) Geometry Project Programs discgrp(5gv)
2
6 discrete group file formats
7
9 An ascii file representation of a discrete geometric group.
10
12 This file format applies to a wide variety of groups encountered in
13 topology and geometry. At the beginning of the file is the string DIS‐
14 CGRP, in conformity with other OOGL file formats (see oogl(5)). The
15 remainder of the file is a series of parenthesized expressions.
16 The file format is based on an expandable set of key markers and val‐
18 ues. A typical entry would be
19 (group borromean )
20 where parentheses enclose the expression. The string 'group' is called
21 the key marker. The remaining text contains a value, in this case a
22 character string which is the name of the group. Key markers currently
23 recognized and their values include:
24 group A name for this group.
25 comment An arbitrary comment [maximum length 256 bytes]
26 [must be enclosed in double quotes]
27 attribute One of a several keyword identifiers including:
28 euclidean is a euclidean group
29 hyperbolic is a hyperbolic group
30 spherical is a spherical group
31 transposed the matrices are transposed
32 finite this is a finite group
33 model In the case that 'hyperbolic' is an attribute, and the dimension is
34 three, this keyword sets the model that is
35 expected for reading subsequent isometries, i.e., expects elements of
36 SL(2,C) if the model is upperhalfspace or conformalball and O(3,1,R) if it is
37 projective.
38 projective default
39 upperhalfspace
40 conformalball [not currently implemented]
41 display These values set characteristics of the the display of the discrete
42 group.
43 centercam keeps the camera centered in central dirichlet domain.
44 zcull removes tiles which lie behind the camera.
45 showcam if centercam is set, displays camgeom, if provided (see below).
46 dimn The dimension of the space.
47 ngens Number of generators.
48 gens List of generators.
49 nels Number of group elements.
50 els List of group elements.
51 wafile Word acceptor file (in DEAGL format version 2.2)
52 camgeom An OOGL-format geometry description to represent the observer.
53 geom An OOGL-format geometry description. Will replace the default Dirichlet domain computation.
54 String-matching of key markers and attribute values is case-insensi‐
56 tive.
57 Currently all matrices must be 4x4 real matrices. Hence the only valid
59 dimensions are 2,3, or 4. The typical case is dimension 3 where there
60 are projective models for hyperbolic, euclidean, and spherical geome‐
61 try. Dimension 2 groups should leave the z-coordinate unchanged.
62 Matrices are considered transposed if they are considered to act on
64 column vectors as in: M.v = v'. This is the usual mathematical conven‐
65 tion convention. So, if you use this convention you must mark your
66 matrices as being transposed.
67 Group elements are specified by an optional character string followed
69 by a list of 16 real numbers. For the case of generators, the charac‐
70 ter string represents the name of that generator. Its main use is with
71 the automatic group software: the name used here should match that of
72 the generator in the word acceptor file. It should, for example, be a
73 single character for the software to work reliably. For composite ele‐
74 ments, the string is the word in the generators which describes this
75 group element.
76 OOGL libraries expect discrete group files to use the '.dgp' suffix.
78 A file which describes the group of the three dimensional torus fol‐
80 lows:
81 DISCGRP
82 (group 3torus )
83 (comment " This is a test of the new discrete group library. " )
84 (attribute Euclidean )
85 (display centercam )
86 (enumdepth 3 ) # look at all words of length 3 or less in generators
87 (enumdist 10.0 ) # and which don't move the origin more than 10.0
88 (dimn 3 )
89 (ngens 3 )
90 (gens
91 a
92 1 0 0 0
93 0 1 0 0
94 0 0 1 0
95 1 0 -0.1 1
96 b
98 1 0 0 0
99 0 1 0 0
100 0 0 1 0
101 0.1 1 0 1
102 c
104 1 0 0 0
105 0 1 0 0
106 0 0 1 0
107 0 0.2 1 1
108 )
110 (cpoint 0.000000 0.000000 0.000000 1.000000 )
111 (camgeom
112 { = OFF
113 5 2 5
114 0 0 0
116 -0.1 0 0.5
117 0.1 0 0.5
118 0 -0.1 0.5
119 0 0.1 0.5
120 3 0 1 2 200 200 0 .8
122 3 0 3 4 0 200 200 .8
123 }
124 )
125
127 Example group files are to be found in the OOGL release in
128 ${GEOM}/data/groups, where e.g., GEOM = /u/gcg/ngrap. See also disc‐
129 grp(3) and automata(1).
130
132 Charlie Gunn.
133
135 For dimension 2, the code should read and write 3 x 3 matrices.
136 Date: 92/08/26 discgrp(5gv)