GAP (HAPcryst) - Chapter 2: Bits and Pieces

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2 Bits and Pieces
2.1 Matrices and Vectors
2.1-1 SignRat
2.1-2 VectorModOne
2.1-3 IsSquareMat
2.1-4 DimensionSquareMat
2.2 Affine Matrices OnRight
2.2-1 LinearPartOfAffineMatOnRight
2.2-2 BasisChangeAffineMatOnRight
2.2-3 TranslationOnRightFromVector
2.3 Geometry
2.3-1 GramianOfAverageScalarProductFromFiniteMatrixGroup
2.3-2 Inequalities
2.3-3 BisectorInequalityFromPointPair
2.3-4 WhichSideOfHyperplane
2.3-5 RelativePositionPointAndPolygon
2.4 Space Groups
2.4-1 PointGroupRepresentatives

### 2 Bits and Pieces

This chapter contains a few very basic functions which are needed for space group calculations and were missing in standard GAP.

#### 2.1 Matrices and Vectors

##### 2.1-1 SignRat

`> SignRat`( x ) | ( method )

Returns: sign of the rational number x (Standard GAP currently only has `SignInt`).

##### 2.1-2 VectorModOne

`> VectorModOne`( v ) | ( method )

Returns: Rational vector of the same length with enties in [0,1)

For a rational vector v, this returns the vector with all entries taken "mod 1".

    gap> SignRat((-4)/(-2));
    1
    gap> SignRat(9/(-2));
    -1
    gap> VectorModOne([1/10,100/9,5/6,6/5]);
    [ 1/10, 1/9, 5/6, 1/5 ]

##### 2.1-3 IsSquareMat

`> IsSquareMat`( matrix ) | ( method )

Returns: `true` if matrix is a square matrix and `false` otherwise.

##### 2.1-4 DimensionSquareMat

`> DimensionSquareMat`( matrix ) | ( method )

Returns: Number of lines in the matrix matrix if it is square and `fail` otherwise

    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

Affine mappings of n dimensional space are often written as a pair (A,v) where A is a linear mapping and v is a vector. GAP represents affine mappings by n+1 times n+1 matrices M which satisfy M_{n+1,n+1}=1 and M_{i,n+1}=0 for all 1<= i <= n.

An affine matrix acts on an n dimensional space which is written as a space of n+1 tuples with n+1st entry 1. Here we give two functions to handle these affine matrices.

#### 2.2 Affine Matrices OnRight

##### 2.2-1 LinearPartOfAffineMatOnRight

`> LinearPartOfAffineMatOnRight`( mat ) | ( method )

Returns: the linear part of the affine matrix mat. That is, everything except for the last row and column.

##### 2.2-2 BasisChangeAffineMatOnRight

`> BasisChangeAffineMatOnRight`( transform, mat ) | ( method )

Returns: affine matrix with same dimensions as mat

A basis change transform of an n dimensional space induces a transformation on affine mappings on this space. If mat is a affine matrix (in particular, it is (n+1)x (n+1)), this method returns the image of mat under the basis transformation induced by transform.

    gap> c:=[[0,1],[1,0]];
    [ [ 0, 1 ], [ 1, 0 ] ]
    gap> m:=[[1/2,0,0],[0,2/3,0],[1,0,1]];
    [ [ 1/2, 0, 0 ], [ 0, 2/3, 0 ], [ 1, 0, 1 ] ]
    gap> BasisChangeAffineMatOnRight(c,m);
    [ [ 2/3, 0, 0 ], [ 0, 1/2, 0 ], [ 0, 1, 1 ] ]

##### 2.2-3 TranslationOnRightFromVector

`> TranslationOnRightFromVector`( v ) | ( method )

Returns: Affine matrix

Given a vector v with n entries, this method returns a (n+1)x (n+1) matrix which corresponds to the affine translation defined by v.

    gap> m:=TranslationOnRightFromVector([1,2,3]);;
    gap> Display(m);
    [ [  1,  0,  0,  0 ],
      [  0,  1,  0,  0 ],
      [  0,  0,  1,  0 ],
      [  1,  2,  3,  1 ] ]
    gap> LinearPartOfAffineMatOnRight(m);
    [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ]
    gap> BasisChangeAffineMatOnRight([[3,2,1],[0,1,0],[0,0,1]],m);
    [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 3, 4, 4, 1 ] ]

#### 2.3 Geometry

##### 2.3-1 GramianOfAverageScalarProductFromFiniteMatrixGroup

`> GramianOfAverageScalarProductFromFiniteMatrixGroup`( G ) | ( method )

Returns: Symmetric positive definite matrix

For a finite matrix group G, the gramian matrix of the average scalar product is returned. This is the sum over all gg^t with gin G (actually it is enough to take a generating set). The group G is orthogonal with respect to the scalar product induced by the returned matrix.

##### 2.3-2 Inequalities

Inequalities are represented in the same way they are represented in polymaking. The vector (v_0,...,v_n) represents the inequality 0<= v_0+v_1 x_1+... + v_n x_n.

##### 2.3-3 BisectorInequalityFromPointPair

`> BisectorInequalityFromPointPair`( v1, v2[, gram] ) | ( method )

Returns: vector of length `Length(v1)+1`

Calculates the inequality defining the half-space containing v1 such that `v1-v2` is perpendicular on the bounding hyperplane. And `(v1-v2)/2` is contained in the bounding hyperplane.
If the matrix gram is given, it is used as the gramian matrix. Otherwiese, the standard scalar product is used. It is not checked if gram is positive definite or symmetric.

##### 2.3-4 WhichSideOfHyperplane

`> WhichSideOfHyperplane`( v, ineq ) | ( method )

`> WhichSideOfHyperplaneNC`( v, ineq ) | ( method )

Returns: -1 (below) 0 (in) or 1 (above).

Let v be a vector of length n and ineq an inequality represented by a vector of length n+1. Then `WhichSideOfHyperplane(v, ineq)` returns 1 if v is a solution of the inequality but not the equation given by ineq, it returns 0 if v is a solution to the equation and -1 if it is not a solution of the inequality ineq.

The NC version does not test the input for correctness.

    gap> BisectorInequalityFromPointPair([0,0],[1,0]);
    [ 1, -2, 0 ]
    gap> ineq:=BisectorInequalityFromPointPair([0,0],[1,0],[[5,4],[4,5]]);
    [ 5, -10, -8 ]
    gap> ineq{[2,3]}*[1/2,0];
    -5
    gap> WhichSideOfHyperplane([0,0],ineq);
    1
    gap> WhichSideOfHyperplane([1/2,0],ineq);
    0

##### 2.3-5 RelativePositionPointAndPolygon

`> RelativePositionPointAndPolygon`( point, poly ) | ( method )

Returns: one of `"VERTEX", "FACET", "OUTSIDE", "INSIDE"`

Let poly be a `PolymakeObject` and point a vector. If point is a vertex of poly, the string `"VERTEX"` is returned. If point lies inside poly, `"INSIDE"` is returned and if it lies in a facet, `"FACET"` is returned and if point does not lie inside poly, the function returns `"OUTSIDE"`.

#### 2.4 Space Groups

##### 2.4-1 PointGroupRepresentatives

`> PointGroupRepresentatives`( group ) | ( attribute )

`> PointGroupRepresentatives`( group ) | ( method )

Returns: list of matrices

Given an `AffineCrystGroupOnLeftOrRight` group, this returns a list of representatives of the point group of group. That is, a system of representatives for the factor group modulo translations. This is an attribute of `AffineCrystGroupOnLeftOrRight`

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