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<div class="chlinkprevnexttop"> <a href="chap0_mj.html">[Top of Book]</a>   <a href="chap0_mj.html#contents">[Contents]</a>    <a href="chap1_mj.html">[Previousp>eturnsthe   the  < classArg<var

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[ 1/10, 1/9, 5/6, 1/5 ]
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<p>Returns: <code class="keyw">true

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<h5>2.2-1 LinearPartOfAffineMatOnRight

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<pReturns the partof affinematrix

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<p><a id="X81F3C49580958FB6" name="X81F3C49580958FB6"></a></p>

<h5>2.2-3 TranslationOnRightFromVector</h5>

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<p><a id="X84A0B0637F269E37" name="X84A0B0637F269E37"></a></p>

<h4>2.3 <span class="Heading">Geometry</span></h4>

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<h531GramianOfAverageScalarProductFromFiniteMatrixGroup>

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<h5>2.3-2 <span class="Heading">Inequalities</span></h5>

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<h5>2.3-3 BisectorInequalityFromPointPair

<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
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1
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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>


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