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<p><a id="X86FBE5B77C2F9442" name="X86FBE5B77C2F9442"></a></p>
<div class="ChapSects"><a href="chap2_mj.html#X86FBE5B77C2F9442">2 <span class="Heading">Bits and Pieces</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.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_mj.html#X7D58A1848182EC26">2.1-1 SignRat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X7C0552BA873515B9">2.1-2 VectorModOne</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X7BB083A57C474F45">2.1-3 IsSquareMat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X78C932A48515EF10">2.1-4 DimensionSquareMat</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.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_mj.html#X838946957FC75C17">2.2-1 LinearPartOfAffineMatOnRight</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X80DD4F2286D49F8D">2.2-2 BasisChangeAffineMatOnRight</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X81F3C49580958FB6">2.2-3 TranslationOnRightFromVector</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.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_mj.html#X7A94DAE679AD73E3">2.3-1 GramianOfAverageScalarProductFromFiniteMatrixGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X866942167802E036">2.3-2 <span class="Heading">Inequalities</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X80C365AA87BDDAFA">2.3-3 BisectorInequalityFromPointPair</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X8790464D86D189F4">2.3-4 WhichSideOfHyperplane</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X83392C417B311B6B">2.3-5 RelativePositionPointAndPolygon</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.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_mj.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">&#an style='color: green'>8227; 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">&#an style='color: green'>8227; 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">&#an style='color: green'>8227; 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">&#an style='color: green'>8227; 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">\(1\leq i \leq n\)</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">&#an style='color: green'>8227; 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">&#an style='color: green'>8227; 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">\((n+1)\times (n+1)\)</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">&#an style='color: green'>8227; 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">\((n+1)\times (n+1)\)</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">&#an style='color: green'>8227; 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">\(g\in G\)</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">\((v_0,\ldots,v_n)\)</span> represents the inequality <span class="SimpleMath">\(0\leq v_0+v_1 x_1+\ldots + v_n x_n\)</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">&#an style='color: green'>8227; 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">&#an style='color: green'>8227; 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">&#an style='color: green'>8227; 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">&#an style='color: green'>8227; 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">&#an style='color: green'>8227; 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">&#an style='color: green'>8227; 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>


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