This chapter contains a few very basic functions which are needed for space
group calculations and were missing in standard ⪆.
<Section><Heading>Matrices and Vectors</Heading>
<ManSection>
<Meth Name="SignRat" Arg="x" />
<Returns> sign of the rational number <Arg>x</Arg> (Standard &GAP;
currently only has <C>SignInt</C>).
</Returns>
</ManSection>
<ManSection>
<Meth Name="VectorModOne" Arg="v"/>
<Returns>Rational vector of the same length with enties in <M>[0,1)</M></Returns>
<Description>
For a rational vector <Arg>v</Arg>, this returns the vector with all entries
taken "mod 1".
</Description>
</ManSection>
<ManSection>
<Meth Name="IsSquareMat" Arg="matrix"/>
<Returns>
<K>true</K> if <Arg>matrix</Arg> is a square matrix and <K>false</K> otherwise.
</Returns>
</ManSection>
<ManSection>
<Meth Name="DimensionSquareMat" Arg="matrix"/>
<Returns>Number of lines in the matrix <Arg>matrix</Arg> if it is square and
<K>fail</K> otherwise
</Returns>
</ManSection>
<Example>
gap> m:=[[1,2,3],[4,5,6],[9,6,12]];
[ [ 1, 2, 3 ], [ 4, 5, 6 ], [ 9, 6, 12 ] ]
gap> IsSquareMat(m);
true
gap> DimensionSquareMat(m);
3
gap> DimensionSquareMat([[1,2],[1,2,3]]);
Error, Matrix is not square called from
</Example>
Affine mappings of <M>n</M> dimensional space are often written as a
pair <M>(A,v)</M> where <M>A</M> is a linear mapping and <M>v</M> is a
vector. &GAP; represents affine mappings by <M>n+1</M> times
<M>n+1</M> matrices <M>M</M> which satisfy <M>M_{{n+1,n+1}}=1</M> and
<M>M_{{i,n+1}}=0</M> for all <M>1\leq i \leq n</M>.<P></P>
An affine matrix acts on an <M>n</M> dimensional space which is
written as a space of <M>n+1</M> tuples with <M>n+1</M>st entry
<M>1</M>. Here we give two functions to handle these affine matrices.
<ManSection>
<Meth Name="LinearPartOfAffineMatOnRight" Arg="mat"/>
<Returns>the linear part of the affine matrix <Arg>mat</Arg>. That
is, everything except for the last row and column.</Returns>
</ManSection>
<ManSection>
<Meth Name="BasisChangeAffineMatOnRight" Arg="transform, mat"/>
<Returns>affine matrix with same dimensions as <Arg>mat</Arg></Returns>
<Description>
A basis change <Arg>transform</Arg> of an <M>n</M> dimensional
space induces a transformation on affine mappings on this space.
If <Arg>mat</Arg> is a affine matrix (in particular, it is
<M>(n+1)\times (n+1)</M>), this method returns the image of
<Arg>mat</Arg> under the basis transformation induced by
<Arg>transform</Arg>.
</Description>
</ManSection>
<ManSection>
<Meth Name="TranslationOnRightFromVector" Arg="v"/>
<Returns>Affine matrix </Returns>
<Description>
Given a vector <Arg>v</Arg> with <M>n</M> entries, this method returns a
<M>(n+1)\times (n+1)</M> matrix which corresponds to the affine translation
defined by <Arg>v</Arg>.
</Description>
</ManSection>
<ManSection>
<Meth Name="GramianOfAverageScalarProductFromFiniteMatrixGroup" Arg="G"/>
<Returns>Symmetric positive definite matrix</Returns>
<Description>
For a finite matrix group <A>G</A>, the gramian matrix of the average scalar
product is returned. This is the sum over all <M>gg^t</M> with <M>g\in G</M>
(actually it is enough to take a generating set).
The group <A>G</A> is orthogonal with respect to the scalar product induced
by the returned matrix.
</Description>
</ManSection>
<Subsection><Heading>Inequalities</Heading>
Inequalities are represented in the same way they are represented in
<Package>polymaking</Package>. The vector <M>(v_0,\ldots,v_n)</M>
represents the inequality <M>0\leq v_0+v_1 x_1+\ldots + v_n x_n</M>.
</Subsection>
<ManSection>
<Meth Name="BisectorInequalityFromPointPair" Arg="v1, v2 [,gram]"/>
<Returns>vector of length <C>Length(v1)+1</C></Returns>
<Description>
Calculates the inequality defining the half-space containing <A>v1</A> such
that <C><A>v1</A>-<A>v2</A></C> is perpendicular on the bounding
hyperplane. And <C>(<A>v1</A>-<A>v2</A>)/2</C> is contained in the bounding
hyperplane.<Br/>
If the matrix <A>gram</A> is given, it is used as the gramian
matrix. Otherwiese, the standard scalar product is used. It is not checked
if <A>gram</A> is positive definite or symmetric.
</Description>
</ManSection>
<ManSection>
<Meth Name="WhichSideOfHyperplane" Arg="v, ineq"/>
<Meth Name="WhichSideOfHyperplaneNC" Arg="v, ineq"/>
<Returns>-1 (below) 0 (in) or 1 (above).</Returns>
<Description>
Let <A>v</A> be a vector of length <M>n</M> and <A>ineq</A> an inequality
represented by a vector of length <M>n+1</M>.
Then <C>WhichSideOfHyperplane(<A>v, ineq</A>)</C> returns 1 if <A>v</A> is
a solution of the inequality but not the equation given by <A>ineq</A>, it
returns 0 if <A>v</A> is a solution to the equation and -1 if it is not a
solution of the inequality <A>ineq</A>.
<P/>
The NC version does not test the input for correctness.
</Description>
</ManSection>
<ManSection><Meth Name="RelativePositionPointAndPolygon" Arg="point, poly"/>
<Returns>one of <C>"VERTEX", "FACET", "OUTSIDE", "INSIDE"</C></Returns>
<Description>
Let <A>poly</A> be a <K>PolymakeObject</K> and <A>point</A> a vector. If
<A>point</A> is a vertex of <A>poly</A>, the string <C>"VERTEX"</C> is
returned. If <A>point</A> lies inside <A>poly</A>, <C>"INSIDE"</C> is
returned and if it lies in a facet, <C>"FACET"</C> is returned and if
<A>point</A> does not lie inside <A>poly</A>, the function returns
<C>"OUTSIDE"</C>.
</Description>
</ManSection>
</Section>
<Section><Heading>Space Groups</Heading>
<ManSection><Attr Name="PointGroupRepresentatives" Arg="group"/>
<Meth Name="PointGroupRepresentatives" Arg="group"/>
<Returns>list of matrices</Returns>
<Description>
Given an <K>AffineCrystGroupOnLeftOrRight</K> <A>group</A>, this returns a
list of representatives of the point group of <A>group</A>. That is, a
system of representatives for the factor group modulo translations. This is
an attribute of <K>AffineCrystGroupOnLeftOrRight</K>
</Description>
</ManSection>
</Section>
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