1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
|
<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en">
<head>
<title>GAP (HAPcryst) - Chapter 2: Bits and Pieces</title>
<meta http-equiv="content-type" content="text/html; charset=UTF-8" />
<meta name="generator" content="GAPDoc2HTML" />
<link rel="stylesheet" type="text/css" href="manual.css" />
<script src="manual.js" type="text/javascript"></script>
<script type="text/javascript">overwriteStyle();</script>
</head>
<body class="chap2" onload="jscontent()">
<div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a> <a href="chap1.html">1</a> <a href="chap2.html">2</a> <a href="chap3.html">3</a> <a href="chap4.html">4</a> <a href="chapBib.html">Bib</a> <a href="chapInd.html">Ind</a> </div>
<div class="chlinkprevnexttop"> <a href="chap0.html">[Top of Book]</a> <a href="chap0.html#contents">[Contents]</a> <a href="chap1.html">[Previous Chapter]</a> <a href="chap3.html">[Next Chapter]</a> </div>
<p id="mathjaxlink" class="pcenter"><a href="chap2_mj.html">[MathJax on]</a></p>
<p><a id="X86FBE5B77C2F9442" name="X86FBE5B77C2F9442"></a></p>
<div class="ChapSects"><a href="chap2.html#X86FBE5B77C2F9442">2 <span class="Heading">Bits and Pieces</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X8019925B8294F5B4">2.1 <span class="Heading">Matrices and Vectors</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7D58A1848182EC26">2.1-1 SignRat</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7C0552BA873515B9">2.1-2 VectorModOne</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7BB083A57C474F45">2.1-3 IsSquareMat</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X78C932A48515EF10">2.1-4 DimensionSquareMat</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X86BD4FE4871379AD">2.2 <span class="Heading">Affine Matrices OnRight</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X838946957FC75C17">2.2-1 LinearPartOfAffineMatOnRight</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X80DD4F2286D49F8D">2.2-2 BasisChangeAffineMatOnRight</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X81F3C49580958FB6">2.2-3 TranslationOnRightFromVector</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X84A0B0637F269E37">2.3 <span class="Heading">Geometry</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7A94DAE679AD73E3">2.3-1 GramianOfAverageScalarProductFromFiniteMatrixGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X866942167802E036">2.3-2 <span class="Heading">Inequalities</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X80C365AA87BDDAFA">2.3-3 BisectorInequalityFromPointPair</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X8790464D86D189F4">2.3-4 WhichSideOfHyperplane</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X83392C417B311B6B">2.3-5 RelativePositionPointAndPolygon</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7B14774981F80108">2.4 <span class="Heading">Space Groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7E2F20607F278B70">2.4-1 PointGroupRepresentatives</a></span>
</div></div>
</div>
<h3>2 <span class="Heading">Bits and Pieces</span></h3>
<p>This chapter contains a few very basic functions which are needed for space group calculations and were missing in standard <strong class="pkg">GAP</strong>.</p>
<p><a id="X8019925B8294F5B4" name="X8019925B8294F5B4"></a></p>
<h4>2.1 <span class="Heading">Matrices and Vectors</span></h4>
<p><a id="X7D58A1848182EC26" name="X7D58A1848182EC26"></a></p>
<h5>2.1-1 SignRat</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SignRat</code>( <var class="Arg">x</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: sign of the rational number <var class="Arg">x</var> (Standard <strong class="pkg">GAP</strong> currently only has <code class="code">SignInt</code>).</p>
<p><a id="X7C0552BA873515B9" name="X7C0552BA873515B9"></a></p>
<h5>2.1-2 VectorModOne</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ VectorModOne</code>( <var class="Arg">v</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: Rational vector of the same length with enties in <span class="SimpleMath">[0,1)</span></p>
<p>For a rational vector <var class="Arg">v</var>, this returns the vector with all entries taken "mod 1".</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">SignRat((-4)/(-2));</span>
1
<span class="GAPprompt">gap></span> <span class="GAPinput">SignRat(9/(-2));</span>
-1
<span class="GAPprompt">gap></span> <span class="GAPinput">VectorModOne([1/10,100/9,5/6,6/5]);</span>
[ 1/10, 1/9, 5/6, 1/5 ]
</pre></div>
<p><a id="X7BB083A57C474F45" name="X7BB083A57C474F45"></a></p>
<h5>2.1-3 IsSquareMat</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSquareMat</code>( <var class="Arg">matrix</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: <code class="keyw">true</code> if <var class="Arg">matrix</var> is a square matrix and <code class="keyw">false</code> otherwise.</p>
<p><a id="X78C932A48515EF10" name="X78C932A48515EF10"></a></p>
<h5>2.1-4 DimensionSquareMat</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DimensionSquareMat</code>( <var class="Arg">matrix</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: Number of lines in the matrix <var class="Arg">matrix</var> if it is square and <code class="keyw">fail</code> otherwise</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">m:=[[1,2,3],[4,5,6],[9,6,12]];</span>
[ [ 1, 2, 3 ], [ 4, 5, 6 ], [ 9, 6, 12 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSquareMat(m);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">DimensionSquareMat(m);</span>
3
<span class="GAPprompt">gap></span> <span class="GAPinput">DimensionSquareMat([[1,2],[1,2,3]]);</span>
Error, Matrix is not square called from
</pre></div>
<p>Affine mappings of <span class="SimpleMath">n</span> dimensional space are often written as a pair <span class="SimpleMath">(A,v)</span> where <span class="SimpleMath">A</span> is a linear mapping and <span class="SimpleMath">v</span> is a vector. <strong class="pkg">GAP</strong> represents affine mappings by <span class="SimpleMath">n+1</span> times <span class="SimpleMath">n+1</span> matrices <span class="SimpleMath">M</span> which satisfy <span class="SimpleMath">M_{n+1,n+1}=1</span> and <span class="SimpleMath">M_{i,n+1}=0</span> for all <span class="SimpleMath">1≤ i ≤ n</span>.</p>
<p>An affine matrix acts on an <span class="SimpleMath">n</span> dimensional space which is written as a space of <span class="SimpleMath">n+1</span> tuples with <span class="SimpleMath">n+1</span>st entry <span class="SimpleMath">1</span>. Here we give two functions to handle these affine matrices.</p>
<p><a id="X86BD4FE4871379AD" name="X86BD4FE4871379AD"></a></p>
<h4>2.2 <span class="Heading">Affine Matrices OnRight</span></h4>
<p><a id="X838946957FC75C17" name="X838946957FC75C17"></a></p>
<h5>2.2-1 LinearPartOfAffineMatOnRight</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LinearPartOfAffineMatOnRight</code>( <var class="Arg">mat</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: the linear part of the affine matrix <var class="Arg">mat</var>. That is, everything except for the last row and column.</p>
<p><a id="X80DD4F2286D49F8D" name="X80DD4F2286D49F8D"></a></p>
<h5>2.2-2 BasisChangeAffineMatOnRight</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BasisChangeAffineMatOnRight</code>( <var class="Arg">transform</var>, <var class="Arg">mat</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: affine matrix with same dimensions as <var class="Arg">mat</var></p>
<p>A basis change <var class="Arg">transform</var> of an <span class="SimpleMath">n</span> dimensional space induces a transformation on affine mappings on this space. If <var class="Arg">mat</var> is a affine matrix (in particular, it is <span class="SimpleMath">(n+1)× (n+1)</span>), this method returns the image of <var class="Arg">mat</var> under the basis transformation induced by <var class="Arg">transform</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">c:=[[0,1],[1,0]];</span>
[ [ 0, 1 ], [ 1, 0 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">m:=[[1/2,0,0],[0,2/3,0],[1,0,1]];</span>
[ [ 1/2, 0, 0 ], [ 0, 2/3, 0 ], [ 1, 0, 1 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">BasisChangeAffineMatOnRight(c,m);</span>
[ [ 2/3, 0, 0 ], [ 0, 1/2, 0 ], [ 0, 1, 1 ] ]
</pre></div>
<p><a id="X81F3C49580958FB6" name="X81F3C49580958FB6"></a></p>
<h5>2.2-3 TranslationOnRightFromVector</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TranslationOnRightFromVector</code>( <var class="Arg">v</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: Affine matrix</p>
<p>Given a vector <var class="Arg">v</var> with <span class="SimpleMath">n</span> entries, this method returns a <span class="SimpleMath">(n+1)× (n+1)</span> matrix which corresponds to the affine translation defined by <var class="Arg">v</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">m:=TranslationOnRightFromVector([1,2,3]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display(m);</span>
[ [ 1, 0, 0, 0 ],
[ 0, 1, 0, 0 ],
[ 0, 0, 1, 0 ],
[ 1, 2, 3, 1 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">LinearPartOfAffineMatOnRight(m);</span>
[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">BasisChangeAffineMatOnRight([[3,2,1],[0,1,0],[0,0,1]],m);</span>
[ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 3, 4, 4, 1 ] ]
</pre></div>
<p><a id="X84A0B0637F269E37" name="X84A0B0637F269E37"></a></p>
<h4>2.3 <span class="Heading">Geometry</span></h4>
<p><a id="X7A94DAE679AD73E3" name="X7A94DAE679AD73E3"></a></p>
<h5>2.3-1 GramianOfAverageScalarProductFromFiniteMatrixGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GramianOfAverageScalarProductFromFiniteMatrixGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: Symmetric positive definite matrix</p>
<p>For a finite matrix group <var class="Arg">G</var>, the gramian matrix of the average scalar product is returned. This is the sum over all <span class="SimpleMath">gg^t</span> with <span class="SimpleMath">g∈ G</span> (actually it is enough to take a generating set). The group <var class="Arg">G</var> is orthogonal with respect to the scalar product induced by the returned matrix.</p>
<p><a id="X866942167802E036" name="X866942167802E036"></a></p>
<h5>2.3-2 <span class="Heading">Inequalities</span></h5>
<p>Inequalities are represented in the same way they are represented in <strong class="pkg">polymaking</strong>. The vector <span class="SimpleMath">(v_0,...,v_n)</span> represents the inequality <span class="SimpleMath">0≤ v_0+v_1 x_1+... + v_n x_n</span>.</p>
<p><a id="X80C365AA87BDDAFA" name="X80C365AA87BDDAFA"></a></p>
<h5>2.3-3 BisectorInequalityFromPointPair</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BisectorInequalityFromPointPair</code>( <var class="Arg">v1</var>, <var class="Arg">v2</var>[, <var class="Arg">gram</var>] )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: vector of length <code class="code">Length(v1)+1</code></p>
<p>Calculates the inequality defining the half-space containing <var class="Arg">v1</var> such that <code class="code"><var class="Arg">v1</var>-<var class="Arg">v2</var></code> is perpendicular on the bounding hyperplane. And <code class="code">(<var class="Arg">v1</var>-<var class="Arg">v2</var>)/2</code> is contained in the bounding hyperplane.<br /> If the matrix <var class="Arg">gram</var> is given, it is used as the gramian matrix. Otherwiese, the standard scalar product is used. It is not checked if <var class="Arg">gram</var> is positive definite or symmetric.</p>
<p><a id="X8790464D86D189F4" name="X8790464D86D189F4"></a></p>
<h5>2.3-4 WhichSideOfHyperplane</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ WhichSideOfHyperplane</code>( <var class="Arg">v</var>, <var class="Arg">ineq</var> )</td><td class="tdright">( method )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ WhichSideOfHyperplaneNC</code>( <var class="Arg">v</var>, <var class="Arg">ineq</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: -1 (below) 0 (in) or 1 (above).</p>
<p>Let <var class="Arg">v</var> be a vector of length <span class="SimpleMath">n</span> and <var class="Arg">ineq</var> an inequality represented by a vector of length <span class="SimpleMath">n+1</span>. Then <code class="code">WhichSideOfHyperplane(<var class="Arg">v, ineq</var>)</code> returns 1 if <var class="Arg">v</var> is a solution of the inequality but not the equation given by <var class="Arg">ineq</var>, it returns 0 if <var class="Arg">v</var> is a solution to the equation and -1 if it is not a solution of the inequality <var class="Arg">ineq</var>.</p>
<p>The NC version does not test the input for correctness.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">BisectorInequalityFromPointPair([0,0],[1,0]);</span>
[ 1, -2, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ineq:=BisectorInequalityFromPointPair([0,0],[1,0],[[5,4],[4,5]]);</span>
[ 5, -10, -8 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ineq{[2,3]}*[1/2,0];</span>
-5
<span class="GAPprompt">gap></span> <span class="GAPinput">WhichSideOfHyperplane([0,0],ineq);</span>
1
<span class="GAPprompt">gap></span> <span class="GAPinput">WhichSideOfHyperplane([1/2,0],ineq);</span>
0
</pre></div>
<p><a id="X83392C417B311B6B" name="X83392C417B311B6B"></a></p>
<h5>2.3-5 RelativePositionPointAndPolygon</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RelativePositionPointAndPolygon</code>( <var class="Arg">point</var>, <var class="Arg">poly</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: one of <code class="code">"VERTEX", "FACET", "OUTSIDE", "INSIDE"</code></p>
<p>Let <var class="Arg">poly</var> be a <code class="keyw">PolymakeObject</code> and <var class="Arg">point</var> a vector. If <var class="Arg">point</var> is a vertex of <var class="Arg">poly</var>, the string <code class="code">"VERTEX"</code> is returned. If <var class="Arg">point</var> lies inside <var class="Arg">poly</var>, <code class="code">"INSIDE"</code> is returned and if it lies in a facet, <code class="code">"FACET"</code> is returned and if <var class="Arg">point</var> does not lie inside <var class="Arg">poly</var>, the function returns <code class="code">"OUTSIDE"</code>.</p>
<p><a id="X7B14774981F80108" name="X7B14774981F80108"></a></p>
<h4>2.4 <span class="Heading">Space Groups</span></h4>
<p><a id="X7E2F20607F278B70" name="X7E2F20607F278B70"></a></p>
<h5>2.4-1 PointGroupRepresentatives</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PointGroupRepresentatives</code>( <var class="Arg">group</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PointGroupRepresentatives</code>( <var class="Arg">group</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: list of matrices</p>
<p>Given an <code class="keyw">AffineCrystGroupOnLeftOrRight</code> <var class="Arg">group</var>, this returns a list of representatives of the point group of <var class="Arg">group</var>. That is, a system of representatives for the factor group modulo translations. This is an attribute of <code class="keyw">AffineCrystGroupOnLeftOrRight</code></p>
<div class="chlinkprevnextbot"> <a href="chap0.html">[Top of Book]</a> <a href="chap0.html#contents">[Contents]</a> <a href="chap1.html">[Previous Chapter]</a> <a href="chap3.html">[Next Chapter]</a> </div>
<div class="chlinkbot"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a> <a href="chap1.html">1</a> <a href="chap2.html">2</a> <a href="chap3.html">3</a> <a href="chap4.html">4</a> <a href="chapBib.html">Bib</a> <a href="chapInd.html">Ind</a> </div>
<hr />
<p class="foot">generated by <a href="http://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc">GAPDoc2HTML</a></p>
</body>
</html>
|
