# application: common

This artificial application gathers functionality shared by many "real" applications. While most users can probably do without looking into this, you may find some useful functions here.

## Objects

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### PermBase

Category: Combinatorics

Base class for permutations of big' objects

#### Properties of PermBase

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PERMUTATION: Array<Int>

Mapping of features stored in this subobject onto their order in the parent (non-permuted) object. The features to be permuted can be any discrete objects with identity imparted by their involvement in a larger structure, like graph nodes, cone rays, facets, complex celles, etc.

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### Visual::Container

Category: Visualization

The common base class of all visual objects composed of several simpler objects. Instances of such classes can carry default decoration attributes applied to all contained objects.

derived from: Visual::Object
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### Visual::Object

Category: Visualization

The common base class of all visualization artifacts produced by various user methods like VISUAL, VISUAL_GRAPH, SCHLEGEL, etc. Visual objects can be passed to functions explicitly calling visualization software like jreality() or povray().

## User Functions

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db_get_type_information ()

20171204: Made type_information_key undef by default to allow users to pass the key "default" explicitly, otherwise default keys have the form "default.<collection>" as there may be more than one collection in the db with a default type information entry

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•

### Arithmetic

These are functions that perform arithmetic computations.

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ceil (a) → Rational

The ceiling function. Returns the smallest integral number not smaller than a.

##### Parameters
 Rational a
##### Returns
 Rational
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denominator (a) → Integer

Returns the denominator of a in a reduced representation.

##### Parameters
 Rational a
##### Returns
 Integer
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denominator (f) → Polynomial

Returns the denominator of a RationalFunction f.

##### Parameters
 RationalFunction f
##### Returns
 Polynomial
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denominator (f) → Polynomial

Returns the denominator of a PuiseuxFraction f.

##### Parameters
 PuiseuxFraction f
##### Returns
 Polynomial
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div (a, b) → Div

Compute the quotient and remainder of a and b in one operation.

##### Parameters
 Int a Int b
##### Returns
 Div

Example:
• > $d = div(10,3);> print$d->quot; 3> print $d->rem; 1 • div_exact (a, b) → Integer Computes the ratio of two given integral numbers under the assumption that the dividend is a multiple of the divisor. ##### Parameters  Integer a Integer b a divisor of a ##### Returns  Integer Example: • > print div_exact(10,5); 2 • ext_gcd (a, b) → ExtGCD Compute the greatest common divisor of two numbers (a,b) and accompanying co-factors. ##### Parameters  Int a Int b ##### Returns  ExtGCD Example: • >$GCD = ext_gcd(15,6); The GCD of the numbers can then be accessed like this:> print $GCD->g; 3 The ExtGCD type also stores the Bezout coefficients (thus integers p and q such that g=a*p+b*q)...> print$GCD->p; 1 print $GCD->q; -2 ...and the quotients k1 of a and k2 of b by g.> print$GCD->k1; 5> print $GCD->k2; 2 • fac (n) → Integer Computes the factorial n! = n·(n-1)·(n-2)·...·2·1. ##### Parameters  Int n >=0 ##### Returns  Integer n! • floor (a) → Rational The floor function. Returns the smallest integral number not larger than a. ##### Parameters  Rational a ##### Returns  Rational Example: • > print floor(1.8); 1 • gcd (a, b) → Int Computes the greatest common divisor of two integers. ##### Parameters  Int a Int b ##### Returns  Int Example: • > print gcd(6,9); 3 • gcd (v) → Element Compute the greatest common divisor of the elements of the given vector. ##### Parameters  Vector v ##### Returns  Element Example: • >$v = new Vector<Int>(3,6,9);> print gcd($v); 3 • gcd (p, q) → UniPolynomial Returns the greatest common divisor of two univariate polynomials. ##### Parameters  UniPolynomial p UniPolynomial q ##### Returns  UniPolynomial Example: • We create two UniPolynomials with said coefficient and exponent type:>$p = new UniPolynomial<Rational,Int>([2,2],[3,2]);> $q = new UniPolynomial<Rational,Int>([6,4],[4,2]); Printing them reveals what the constructor does:> print$p; 2*x^3 + 2*x^2> print $q; 6*x^4 + 4*x^2 Now we can calculate their gcd:> print gcd($p,$q); x^2 • get_var_names () → Array<String> Get the current list of variable names used for pretty printing and string parsing of the given polynomial class ##### Returns  Array • isfinite (a) → Bool Check whether the given number has a finite value. ##### Parameters  SCALAR a ##### Returns  Bool Example: • > print isfinite('inf'); false> print isfinite(23); true • isinf (a) → Int Check whether the given number has an infinite value. Return -1/+1 for infinity and 0 for all finite values. ##### Parameters  SCALAR a ##### Returns  Int Example: • > print isinf('inf'); 1> print isinf(23); 0 • is_one (s) → Bool Compare with the one (1) value of the corresponding data type. ##### Parameters  SCALAR s ##### Returns  Bool • is_zero (s) → Bool Compare with the zero (0) value of the corresponding data type. ##### Parameters  SCALAR s ##### Returns  Bool • lcm (a, b) → Int Computes the least common multiple of two integers. ##### Parameters  Int a Int b ##### Returns  Int Example: • > print lcm(6,9); 18 • lcm (v) → Element Compute the least common multiple of the elements of the given vector. ##### Parameters  Vector v ##### Returns  Element Example: • >$v = new Vector<Integer>(1,3,6,9);> print lcm($v); 18 • local_var_names (names ...) Set the list of variable names for given polynomial class temporarily. The existing name list or the default scheme is restored at the end of the current user cycle, similarly to core::prefer_now. ##### Parameters  String names ... variable names, see set_var_names. • monomials <Coefficient, Exponent> (n) → UniPolynomial<Coefficient,Exponent> Create degree one monomials of the desired polynomial type. ##### Type Parameters  Coefficient The polynomial coefficient type. Rational by default. Exponent The exponent type. Int by default. ##### Parameters  Int n The number of variables ##### Returns  UniPolynomial when n == 1, Polynomial when n > 1 • numerator (a) → Integer Returns the numerator of a in a reduced representation. ##### Parameters  Rational a ##### Returns  Integer • numerator (f) → Polynomial Returns the numerator of a RationalFunction f. ##### Parameters  RationalFunction f ##### Returns  Polynomial • numerator (f) → Polynomial Returns the numerator of a PuiseuxFraction f. ##### Parameters  PuiseuxFraction f ##### Returns  Polynomial • set_var_names (names ...) Set the list of variable names used for pretty printing and string parsing of the given polynomial class When the number of variables in a polynomial is greater than the size of the name list, the excess variable names are produced from a template "${last_var_name}_{EXCESS}", where EXCESS starts at 0 for the variable corresponding to the last name in the list. If the last name already has a form "{Name}_{Number}", the following variables are enumerated starting from that Number plus 1.

The default naming scheme consists of a single letter "x", "y", "z", "u", "v", or "w" chosen according to the nesting depth of polynomial types in the coefficient type. That is, variables of simple polynomials (those with pure numerical coefficients) are named x_0, x_1, ..., variables of polynomials with simple polynomial coefficients are named y_0, y_1, etc.

##### Parameters
 String names ... variable names; may also be bundled in an array an empty list resets to the default naming scheme
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sum_of_square_roots_naive (input_array) → Map<Rational, Rational>

Make a naive attempt to sum the square roots of the entries of the input array.

##### Parameters
 Array input_array a list of rational numbers (other coefficents are not implemented).
##### Returns
 Map a map collecting the coefficients of roots encountered in the sum.

Example:
• To obtain sqrt{3/4} + sqrt{245}, type> print sum_of_square_roots_naive(new Array<Rational>([3/4, 245])); {(3 1/2) (5 7)} This output represents sqrt{3}/2 + 7 sqrt{5}. If you are not satisfied with the result, please use a symbolic algebra package.
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### Combinatorics

This category contains combinatorial functions.

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all_permutations (n) → ARRAY

Returns a list of all permutations of the set {0...n-1} as a perl-array

##### Parameters
 Int n
##### Returns
 ARRAY

Example:
• To store the result in the perl array @a, type this:> @a = all_permutations(3); The array contains pointers to arrays. To access the 0-th pointer, do this:> $a0 =$a[0]; To print the 0-th array itself, you have to dereference it as follows:> print @{ $a0 }; 012 You can loop through @a using foreach. The print statement produces the string obtained by dereferencing the current entry concatenated with the string " ".> foreach( @a ){> print @{$_ }, " ";> } 012 102 201 021 120 210
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are_permuted (a, b) → Bool

Determine whether two arrays a and b are permuted copies of each other.

##### Parameters
 Array a Array b
##### Returns
 Bool

Example:
• > print are_permuted([1,8,3,4],[3,8,4,1]); true
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binomial (n, k) → Int

Computes the binomial coefficient n choose k. Negative values of n (and k) are supported.

##### Parameters
 Int n Int k
##### Returns
 Int n choose k

Example:
• Print 6 choose 4 like this:> print binomial(6,4); 15
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find_permutation (a, b) → Array<Int>

Returns the permutation that maps a to b.

##### Parameters
 Array a Array b
##### Returns
 Array

Example:
• > $p = find_permutation([1,8,3,4],[3,8,4,1]);> print$p; 2 1 3 0
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n_fixed_points (p) → Int

Returns the number of fixed points of the permutation given by p.

##### Parameters
 Array p
##### Returns
 Int

Example:
• > print n_fixed_points([1,0,2,4,3]); 1
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permutation_cycles (p) → ARRAY

Returns the cycles of a permutation given by p.

##### Parameters
 Array p
##### Returns
 ARRAY

Example:
• > print permutation_cycles([1,0,3,2]); {0 1}{2 3}
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permutation_cycle_lengths (p) → Array<Int>

Returns the sorted cycle lengths of a permutation

##### Parameters
 Array p
##### Returns
 Array

Example:
• > print permutation_cycle_lengths(new Array<Int>([1,2,0,4,3])); 2 3
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permutation_matrix <Scalar> (p) → Matrix<Scalar>

Returns the permutation matrix of the permutation given by p.

##### Type Parameters
 Scalar default: Int
##### Parameters
 Array p
##### Returns
 Matrix

Example:
• The following prints the permutation matrix in sparse representation.> print permutation_matrix([1,0,3,2]); (4) (1 1) (4) (0 1) (4) (3 1) (4) (2 1)
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permutation_order (p) → Int

Returns the order of a permutation

##### Parameters
 Array p
##### Returns
 Int

Example:
• > print permutation_order(new Array<Int>([1,2,0,4,3])); 6
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permutation_sign (p) → Int

Returns the sign of the permutation given by p.

##### Parameters
 Array p
##### Returns
 Int +1 or -1

Example:
• > print permutation_sign([1,0,3,2]); 1
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### Data Conversion

This contains functions for data conversions and type casts.

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cast <Target> (object) → Object

Change the type of the polymake object to one of its base types (aka ancestor in the inheritance hierarchy). The object loses all properties that are unknown in the target type.

##### Type Parameters
 Target the desired new type
##### Parameters
 Object object to be modified
##### Returns
 Object the same object, but with modified type
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cols ()

UNDOCUMENTED

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cols (A) → Container<Vector>

Returns an array containing the columns of A.

##### Parameters
 Matrix A
##### Returns
 Container

Example:
• The following saves the columns of the vertex matrix of a square in the variable $w and then prints its contents using a foreach loop and concatenating each entry with the string " ".>$w = cols(polytope::cube(2)->VERTICES);> foreach( @$w ){> print @{$_}, " ";> } 1111 -11-11 -1-111
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concat_rows (A) → Vector

Concatenates the rows of A.

##### Parameters
 Matrix A
##### Returns
 Vector

Examples:
• Make a vector out of the rows of the vertex matrix of a cube:> $v = concat_rows(polytope::cube(2)->VERTICES);> print$v; 1 -1 -1 1 1 -1 1 -1 1 1 1 1
• For a sparse matrix, the resulting vector is sparse, too.> $vs = concat_rows(unit_matrix(3));> print$vs; (9) (0 1) (4 1) (8 1)
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convert_to <Target> (s) → Target

Explicit conversion to different scalar type.

##### Type Parameters
 Target
##### Parameters
 SCALAR s
##### Returns
 Target
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convert_to <Target> (v) → Vector<Target>

Explicit conversion to a different element type.

##### Type Parameters
 Target
##### Parameters
 Vector v
##### Returns
 Vector

Example:
• > $v = new Vector<Rational>(1/2,2/3,3/4);>$vf = convert_to<Float>($v);> print$vf; 0.5 0.6666666667 0.75
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convert_to <Target> (m) → Matrix<Target>

Explicit conversion to a different element type.

##### Type Parameters
 Target
##### Parameters
 Matrix m
##### Returns
 Matrix

Example:
• > $M = new Matrix<Rational>([1/2,2],[3,2/3]);>$Mf = convert_to<Float>($M);> print$Mf; 0.5 2 3 0.6666666667
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convert_to <Target> (m) → Polynomial<Target>

Explicit conversion to a different coefficient type.

##### Type Parameters
 Target
##### Parameters
 Polynomial m
##### Returns
 Polynomial
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convert_to <Target> (m) → UniPolynomial<Target>

Explicit conversion to a different coefficient type.

##### Type Parameters
 Target
##### Parameters
 UniPolynomial m
##### Returns
 UniPolynomial
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dense (v) → Vector

Return the input vector (which is already in dense form).

##### Parameters
 Vector v
##### Returns
 Vector
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dense (m) → Matrix

Return the input matrix (which is already in dense form).

##### Parameters
 Matrix m
##### Returns
 Matrix
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dense <Element> (v) → Vector<Element>

Convert to an equivalent dense vector of the same element type.

##### Type Parameters
 Element
##### Parameters
 SparseVector v
##### Returns
 Vector
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dense <Element> (m) → Matrix<Element>

Convert to an equivalent dense matrix of the same element type.

##### Type Parameters
 Element
##### Parameters
 SparseMatrix m
##### Returns
 Matrix
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dense (m) → Matrix<Int>

Convert to a dense 0/1 matrix.

##### Parameters
 IncidenceMatrix m
##### Returns
 Matrix
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dense (s, dim) → Vector<Int>

Convert to a dense 0/1 vector of a given dimension.

##### Parameters
 Set s Int dim
##### Returns
 Vector
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index_matrix (m) → IncidenceMatrix

Get the positions of non-zero entries of a sparse matrix.

##### Parameters
 SparseMatrix m
##### Returns
 IncidenceMatrix

Example:
• > $S = new SparseMatrix([1,2,0,0,0,0],[0,0,5,0,0,32]);> print index_matrix($S); {0 1} {2 5}
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indices (v) → Set<Int>

Get the positions of non-zero entries of a sparse vector.

##### Parameters
 SparseVector v
##### Returns
 Set

Example:
• > $v = new SparseVector(0,1,1,0,0,0,2,0,3);> print indices($v); {1 2 6 8}
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lex_ordered (f) → PowerSet<Int>

Visit the facets of f sorted lexicographically.

##### Parameters
 FacetList f
##### Returns
 PowerSet

Example:
• > $f = new FacetList(polytope::cube(2)->VERTICES_IN_FACETS);> print lex_ordered($f); {{0 1} {0 2} {1 3} {2 3}}
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repeat_col (v, i)

Create a Matrix by repeating the given Vector as cols.

##### Parameters
 Vector v Int i

Example:
• > $v = new Vector(23,42,666);>$M = repeat_col($v,3);> print$M; 23 23 23 42 42 42 666 666 666
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repeat_row (v, i)

Create a Matrix by repeating the given Vector as rows.

##### Parameters
 Vector v Int i

Example:
• > $v = new Vector(23,42,666);>$M = repeat_row($v,3);> print$M; 23 42 666 23 42 666 23 42 666
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rows ()

UNDOCUMENTED

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rows (A) → Container<Vector>

Returns an array containing the rows of A.

##### Parameters
 Matrix A
##### Returns
 Container

Example:
• The following saves the rows of the vertex matrix of a square in the variable $w and then prints its contents using a foreach loop and concatenating each entry with the string " ".>$w = rows(polytope::cube(2)->VERTICES);> foreach( @$w ){> print @{$_}, " ";> } 1-1-1 11-1 1-11 111
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support (v) → Set<Int>

Get the positions of non-zero entries of a vector.

##### Parameters
 Vector v
##### Returns
 Set

Example:
• > print support(new Vector(0,23,0,0,23,0,23,0,0,23)); {1 4 6 9}
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toMatrix <Scalar> (A) → SparseMatrix<Scalar>

Convert an IncidenceMatrix to a SparseMatrix.

##### Type Parameters
 Scalar
##### Parameters
 IncidenceMatrix A
##### Returns
 SparseMatrix

Example:
• > $M = toMatrix<Int>(polytope::cube(2)->VERTICES_IN_FACETS);> print$M->type->full_name; SparseMatrix<Int, NonSymmetric>
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toTropicalPolynomial (s, vars) → Polynomial<TropicalNumber<Addition,Rational> >

This converts a string into a tropical polynomial. The syntax for the string is as follows: It is of the form "min(...)" or "max(...)" or "min{...}" or "max{...}", where ... is a comma-separated list of sums of the form "a + bx + c + dy + ...", where a,c are rational numbers, b,d are Ints and x,y are variables. Such a sum can contain several such terms for the same variable and they need not be in any order. Any text that starts with a letter and does not contain any of +-*,(){} or whitespace can be a variable. A term in a sum can be of the form "3x", "3*x", but "x3"will be interpreted as 1 * "x3". Coefficients should not contain letters and there is no evaluation of arithmetic, i.e. "(2+4)*x" does not work (though "2x+4x" would be fine). In fact, further brackets should only be used (but are not necessary!) for single coefficienst, e.g. "(-3)*x". Warning: The parser will remove all brackets before parsing the individual sums. If no further arguments are given, the function will take the number of occuring variables as total number of variables and create a ring for the result. The variables will be sorted alphabetically.

##### Parameters
 String s The string to be parsed String vars Optional. A list of variables. If this is given, all variables used in s must match one of the variables in this list.
##### Returns
 Polynomial > , where Addition depends on whether min or max was used in the string.
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toVector <Scalar> (S, d) → SparseVector<Scalar>

Create a sparse vector having 1's at positions contained in the given set

##### Type Parameters
 Scalar type of apparent 1's
##### Parameters
 Set S Int d dimension of the result
##### Returns
 SparseVector
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vector2col (v) → Matrix

Convert a Vector to a Matrix with a single column.

##### Parameters
 Vector v
##### Returns
 Matrix

Example:
•

### Database

Here you can find the functions to access the polymake database.

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db_count (query) → Int

Returns the number of objects in the database db in collection that match the query.

##### Parameters
 HASH query
##### Options
 String db name of the database, see here or db_info for available databases String collection name of the collection, see here or db_info for available collections Bool local set to 1 if you want to use a local database (on localhost), default 0 String username Some databases might have access control. String password Some databases might have access control. MongoClient client
##### Returns
 Int
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db_cursor (query) → DBCursor

Returns a cursor on the entries for the database db in collection that match the query.

##### Parameters
 HASH query database query
##### Options
 String db database name String collection collection name Int skip skip the first elements, default: 0 Int limit limit the number of objects that will be returned (default: no limit) HASH sort_by sorting of the entries, default by _id String username String password MongoClient client Bool local set to 1 if you want to use a local database (on localhost), default 0
##### Returns
 DBCursor
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db_get_list_col_for_db () → Array

Returns a list of available collections in a database.

##### Options
 String db name of the database Bool local set to 1 if you want to use a local database (on localhost), default 0 String username necessary if reading a database with access control
##### Returns
 Array
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db_get_list_db () → Array

Returns a list of available databases.

##### Options
 Bool local set to 1 if you want to use a local database (on localhost), default 0 String username necessary if reading a database with access control
##### Returns
 Array
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db_get_list_db_col () → Array

Returns a list of available databases and collections (in the form db.collection).

##### Options
 String db name of the database, default: all available databases String collection name of the collection, default: all available collections Bool local set to 1 if you want to use a local database (on localhost), default 0 String username necessary if reading a database with access control
##### Returns
 Array
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db_ids (query) → Array<String>

Returns the IDs of all objects in the database db in collection that match the query. This is only recommended for a reasonably small number of matching objects. If you expect many such objects you should instead construct a DBCursor.

##### Parameters
 HASH query
##### Options
 String db name of the database, see here or db_info for available databases String collection name of the collection, see here or db_info for available collections Bool local set to 1 if you want to use a local database (on localhost), default 0 Int limit limit the number of objects that will be returned (default: no limit) HASH sort_by specify a sorting order Int skip skip the first elements, default: 0 String username Some databases might have access control. String password Some databases might have access control. MongoClient client
##### Returns
 Array
•
db_info ()

Print information about available databases and collections.

##### Options
 String db name of the database, default: all available databases String collection name of the collection, default: all available collections Bool local set to 1 if you want to use a local database (on localhost), default 0 String username necessary if reading a database with access control
•

Retrieve the metadata of an object

##### Parameters
 Object p the polyDB object
##### Returns
•

print the metadata of an object

##### Parameters
 Object p the polyDB object
•
db_query (query) → Array<Core::Object>

Returns all objects in the database db in collection that match the query. This is only recommended for a reasonably small number of matching objects. If you expect many such objects you should instead use a database cursor.

##### Parameters
 HASH query
##### Options
 String db name of the database, see here or db_info for available databases String collection name of the collection, see here or db_info for available collections Bool local set to 1 if you want to use a local database (on localhost), default 0 Int limit limit the number of objects that will be returned (default: no limit) String username Some databases might have access control. String password Some databases might have access control. MongoClient client Int skip skip the first elements, default: 0 HASH sort_by specify a sorting order
##### Returns
 Array
•
db_searchable_fields (db, collection) → Array<String>

Return the list of property names that can be searched in the database for a given database db, collection col and optional template key.

##### Parameters
 String db name of the database, see here or db_info for available databases String collection name of the collection, see here or db_info for available collections
##### Options
 Bool local set to 1 if you want to use a local database (on localhost), default 0
##### Returns
 Array
•
db_write_collection_info ()

Add documentation for a collection You need write access for this.

##### Options
 String db name of the database the description applies to String collection name of the collection the description applies to Bool local set to 1 if you want to use a local database (on localhost), default 0 Bool replace whether an existing documentation should be updated or replaced String file a file with the documentation in json format, only one of file and documentation is allowed HASH documentation a perl hash with the documentation, only one of documentation and file is allowed String username String password String polydb_version version number Bool verbose verbose mode
•
db_write_db_info (db)

Add a db documentation. You need write access for this.

##### Parameters
 String db name of the database the description applies to
##### Options
 Bool local set to 1 if you want to use a local database (on localhost), default 0 Bool replace whether an existing documentation should be updated or replaced String file a file with the documentation in json format, only one of file and documentation is allowed HASH documentation a perl hash with the documentation, only one of documentation and file is allowed String username String password String polydb_version version number Bool verbose verbose mode
•

This category also contains functions that I want to hide from the public because they are not yet completely presentable.

•
db_set_type_information ()

Set or update type (and template) information for collection col in the database db.

Note that you need write access to the type database for this.

##### Options
 String db database name String col collection name String key the key of the type information Bool replace type information for this database, collection and key already exists and should be replaced String file a filename containing the type information as json document (only one of file and type_information allowed) HASH type_information the type information as per hash (only one of file and type_information allowed) String username String password String polyDB_version the version of polyDB this type information applies to Bool verbose
•

### Database Write Access

These are the functions to insert and update objects. You need write access to the database for these.

•
db_insert (obj) → String

Adds an object obj to the collection col in the database db.

##### Parameters
 Core::Object obj
##### Options
 String db name of the database String collection name of the collection String id unique identifier, this is set to the name of the object, a different name should only be used in exceptional cases. This option is only valid if a single object is added. String contributor set the contributor Bool local set to 1 if you want to use a local database (on localhost), default 0 String username String password Bool use_type_information set to 1 to match properties with a template object HASH type_information object template specifying the properties to store String type_information_key template name to load from the database Bool nonew set to 1 if you don't want to compute missing properties but only delete surplus ones (only takes effect if use_type_info is also set to 1) Bool keep_all_props keep all properties present in the polymake object in the stored xml. Does not effect representation in the db. For this, provide a type information without a property mask Bool noinsert dont insert anything, just try evaluate the template MongoClient client
##### Returns
 String
•
db_remove (id) → String

Removes the object with a given id from the collection col in the database db.

##### Parameters
 String id
##### Options
 String db name of the database String collection name of the collection Bool local set to 1 if you want to use a local database (on localhost), default 0 String username String password MongoClient client
##### Returns
 String
•

### Formatting

Functions for pretty printing, labels or latex output of polymake types.

•
labeled (data, elem_labels) → String

Prepares a vector for printing, prepends each element with a label and a colon.

##### Parameters
 Vector data to be printed Array elem_labels optional labels for elements; if data is a Set, or similar, each element will be replaced by its label.
##### Returns
 String

Example:
• > $v = new Vector(0,1,2);> print labeled($v,["zeroth","first","second"]); zeroth:0 first:1 second:2
•
latex (data, elem_labels) → String

LaTeX output of a matrix.

##### Parameters
 Matrix data to be printed Array elem_labels optional labels for elements; if data is an IncidenceMatrix, Array, or similar, each element will be replaced by its label.
##### Returns
 String to be used with \usepackage{amsmath}
•
numbered (data) → String

Equivalent to labeled with omitted elem_labels argument.

##### Parameters
 Vector data to be printed
##### Returns
 String

Example:
• > $data = new Vector(23,42,666);> print numbered($data); 0:23 1:42 2:666
•
print_constraints (M)

Write the rows of a matrix M as inequalities (equations=0) or equations (equations=1) in a readable way. It is possible to specify labels for the coordinates via an optional array coord_labels.

##### Parameters
 Matrix M the matrix whose rows are to be written
##### Options
 Array coord_labels changes the labels of the coordinates Array row_labels changes the labels of the rows Bool homogeneous false if the first coordinate should be interpreted as right hand side Bool equations true if the rows represent equations instead of inequalities

Example:
• > $M = new Matrix([1,2,3],[4,5,23]);> print_constraints($M,equations=>1); 0: 2 x1 + 3 x2 = -1 1: 5 x1 + 23 x2 = -4
•
rows_labeled (data, row_labels, elem_labels) → Array<String>

Prepares a matrix for printing, prepends each row with a label and a colon.

##### Parameters
 Matrix data to be printed Array row_labels labels for the rows Array elem_labels optional labels for elements; if data is an IncidenceMatrix, Array, or similar, each element will be replaced by its label.
##### Returns
 Array each string ending with end-of-line

Example:
• > print rows_labeled(polytope::cube(2)->VERTICES,['a','b','c','d']); a:1 -1 -1 b:1 1 -1 c:1 -1 1 d:1 1 1
•
rows_labeled (graph, elem_labels) → Array<String>

Like above, but specialized for Graphs (defined for convenience: a PTL Graph is not a container)

##### Parameters
 Graph graph to be printed Array elem_labels labels for the elements
##### Returns
 Array each string ending with end-of-line

Example:
• > print rows_labeled(graph::cycle_graph(4)->ADJACENCY, ['a','b','c','d']); a:b d b:a c c:b d d:a c
•
rows_numbered (data) → Array<String>

Equivalent to rows_labeled with omitted row_labels argument. Formerly called "numbered".

##### Parameters
 Matrix data to be printed
##### Returns
 Array each string ending with end-of-line

Example:
• > print rows_numbered(polytope::cube(2)->VERTICES); 0:1 -1 -1 1:1 1 -1 2:1 -1 1 3:1 1 1
•

### Graph Operations

Operations on graphs.

•

Returns the adjacency matrix of graph nodes. For a normal graph, it will be a kind of IncidenceMatrix, for multigraph, it will be a SparseMatrix<Int>, with entries encoding the number of parallel edges between two nodes.

##### Parameters
 Graph graph
##### Returns
 IncidenceMatrix both rows and columns correspond to the nodes
•
edges (graph) → EdgeList

Returns the sequence of all edges of a graph. The edges will appear in ascending order of their tail and head nodes. In the Undirected case, the edges will appear once, ordered by the larger index of their incident nodes.

##### Parameters
 Graph graph
##### Returns
 EdgeList
•
induced_subgraph (graph, set) → Graph

Creates an induced subgraph for the given subset of nodes.

##### Parameters
 Graph graph Set set indices of selected nodes
##### Returns
 Graph

Example:
• > $g = new props::Graph(graph::cycle_graph(5)->ADJACENCY);>$s1 = new Set(1,2,3);> print induced_subgraph($g,$s1); (5) (1 {2}) (2 {1 3}) (3 {2})
•
nodes (graph) → Set<Int>

Returns the sequence of all valid nodes of a graph.

##### Parameters
 Graph graph
##### Returns
 Set

Example:
• > print nodes(graph::cycle_graph(5)->ADJACENCY); {0 1 2 3 4}
•
node_edge_incidences <Coord> (graph) → SparseMatrix<Coord>

Returns the node-edge incidence matrix of a graph.

##### Type Parameters
 Coord coordinate type for the resulting matrix, default: Int
##### Parameters
 Graph graph
##### Returns
 SparseMatrix

Example:
• > print node_edge_incidences(graph::cycle_graph(5)->ADJACENCY); (5) (0 1) (3 1) (5) (0 1) (1 1) (5) (1 1) (2 1) (5) (2 1) (4 1) (5) (3 1) (4 1)
•

### Lattice Tools

Functions for lattice related computations.

•
eliminate_denominators (v) → Vector<Integer>

Scale a vector with the least common multiple of the denominators of its coordinates.

##### Parameters
 Vector v
##### Returns
 Vector

Example:
• > $v = new Vector(1/2,1/3,1/4,1/5);>$ve = eliminate_denominators($v);> print$ve; 30 20 15 12
•
eliminate_denominators_entire (v) → Matrix<Integer>

Scales entire matrix with the least common multiple of the denominators of its coordinates.

##### Parameters
 Matrix v
##### Returns
 Matrix

Example:
• > $M = new Matrix([1/2,1/3],[1/5,7],[1/4,4/3]);>$Me = eliminate_denominators_entire($M);> print$Me; 30 20 12 420 15 80
•
eliminate_denominators_entire_affine (v) → Matrix<Integer>

Scales entire matrix with the least common multiple of the denominators of its coordinates (ignore first column).

##### Parameters
 Matrix v
##### Returns
 Matrix

Example:
• > $M = new Matrix([1,1/2,1/3],[1,1/5,7],[1,1/4,4/3]);>$Me = eliminate_denominators_entire_affine($M);> print$Me; 1 30 20 1 12 420 1 15 80
•
eliminate_denominators_in_rows (M) → Matrix<Integer>

Scale a matrix row-wise with the least common multiple of the denominators of its coordinates.

##### Parameters
 Matrix M
##### Returns
 Matrix

Example:
• > $M = new Matrix([1/2,1/3],[1/5,7],[1/4,4/3]);>$Me = eliminate_denominators_in_rows($M);> print$Me; 3 2 1 35 3 16
•
is_integral (v) → Bool

Checks whether all coordinates of a rational vector are integral.

##### Parameters
 Vector v
##### Returns
 Bool

Example:
• This rational vector has only integral entries:> $v = new Vector<Rational>(1,2,3,4); polytope > print is_integral($v); true But if we append 1/2, it isn't anymore:> print is_integral($v|1/2); false • is_integral (m) → Bool Checks whether all coordinates of a rational matrix are integral. ##### Parameters  Matrix m ##### Returns  Bool Example: • This rational matrix has only integral entries:>$m = new Matrix<Rational>([1,2],[3,4]);> print is_integral($m); true But if we multiply it with 1/2, that is not the case anymore.> print is_integral(1/2 *$m); false
•
primitive (v) → Vector<Integer>

Scales the vector to a primitive integral vector.

##### Parameters
 Vector v
##### Returns
 Vector

Example:
• > print primitive(new Vector(3,3/2,3,3)); 2 1 2 2
•
primitive (M) → Matrix<Integer>

Scales each row of the matrix to a primitive integral vector.

##### Parameters
 Matrix M
##### Returns
 Matrix

Example:
• > print primitive(new Matrix([1,3/2],[3,1])); 2 3 3 1
•
primitive_affine (v) → Vector<Integer>

Scales the affine part of a vector to a primitive integral vector.

##### Parameters
 Vector v
##### Returns
 Vector

Example:
• > print primitive_affine(new Vector(1,3/2,1,1)); 1 3 2 2
•
primitive_affine (M) → Matrix<Integer>

Scales the affine part of each row of the matrix to a primitive integral vector.

##### Parameters
 Matrix M
##### Returns
 Matrix

Example:
• > print primitive_affine(new Matrix([1,1,3/2],[1,3,1])); 1 2 3 1 3 1
•

### Linear Algebra

These functions are for algebraic computations and constructions of special matrices.

•
anti_diag (d) → SparseMatrix

Produces a SparseMatrix from its anti-diagonal.

##### Parameters
 Vector d the anti-diagonal entries
##### Returns
 SparseMatrix

Example:
• Try this:> $M = anti_diag(new Vector([0,1,2]));> print$M; (3) (2 2) (3) (1 1) (3) To print a more human-readable representation, use the dense() function:> print dense($M); 0 0 2 0 1 0 0 0 0 • anti_diag (m1, m2) → SparseMatrix Returns a block anti-diagonal matrix with blocks m1 and m2. ##### Parameters  Matrix m1 Matrix m2 ##### Returns  SparseMatrix Example: • Try this:>$M = anti_diag(unit_matrix(2),unit_matrix(3));> print $M; (5) (2 1) (5) (3 1) (5) (4 1) (5) (0 1) (5) (1 1) • barycenter (A) → Vector<Scalar> Calculate the average over the rows of a matrix. ##### Parameters  Matrix A ##### Returns  Vector Example: • >$A = new Matrix([3,0,0],[0,3,0],[0,0,3]);> print barycenter($A); 1 1 1 • basis (A) → Pair<Set<Int>, Set<Int>> Computes subsets of the rows and columns of A that form a basis for the linear space spanned by A. ##### Parameters  Matrix A ##### Returns  Pair, Set> The first set corresponds to the rows, the second to the columns. Example: • Here we have a nice matrix:>$M = new Matrix([[1,0,0,0],[2,0,0,0],[0,1,0,0],[0,0,1,0]]); Let's print bases for the row and column space:> ($row,$col) = basis($M);> print$M->minor($row,All); 1 0 0 0 0 1 0 0 0 0 1 0> print$M->minor(All,$col); 1 0 0 2 0 0 0 1 0 0 0 1 • basis_affine (A) → Pair<Set<Int>, Set<Int>> Does the same as basis ignoring the first column of the matrix. ##### Parameters  Matrix A ##### Returns  Pair, Set> The first set corresponds to the rows, the second to the columns. • basis_cols (A) → Set<Int> Computes a subset of the columns of A that form a basis for the linear space spanned by A. ##### Parameters  Matrix A ##### Returns  Set Example: • Here we have a nice matrix:>$M = new Matrix([[1,0,0,0],[2,0,0,0],[0,1,0,0],[0,0,1,0]]); Let's print a basis of its column space:> print $M->minor(All,basis_cols($M)); 1 0 0 2 0 0 0 1 0 0 0 1
•
basis_rows (A) → Set<Int>

Computes a subset of the rows of A that form a basis for the linear space spanned by A.

##### Parameters
 Matrix A
##### Returns
 Set

Example:
• Here we have a nice matrix:> $M = new Matrix([[1,0,0,0],[2,0,0,0],[0,1,0,0],[0,0,1,0]]); Let's print a basis of its row space:> print$M->minor(basis_rows($M),All); 1 0 0 0 0 1 0 0 0 0 1 0 • basis_rows_integer (A) → Set<Int> Computes a lattice basis of the span of the rows of A. ##### Parameters  Matrix A ##### Returns  Set Indices of the rows of A that constitute a basis. • cramer (A, b) → Vector Computes the solution of the system Ax = b by applying Cramer's rule ##### Parameters  Matrix A must be invertible Vector b ##### Returns  Vector Example: • from the Wikipedia:>$A = new Matrix([3,2,-1],[2,-2,4],[-1,1/2,-1]);> $b = new Vector(1,-2,0);> print cramer($A,$b); 1 -2 -2 • det (A) → Scalar Computes the determinant of a matrix using Gaussian elimination. If Scalar is not of field type, but element of a Euclidean ring R, type upgrade to element of the quotient field is performed. The result is recast as a Scalar, which is possible without roundoff since the so-computed determinant is an element of the (embedded) ring R. ##### Parameters  Matrix A ##### Returns  Scalar det(A) Examples: • > print det(unit_matrix(3)); 1 • >$p = new UniPolynomial<Rational,Int>("x2+3x");> $M = new Matrix<UniPolynomial<Rational,Int>>([[$p, $p+1],[$p+1,$p]]);> print det($M); -2*x^2 -6*x - 1
•
diag (d) → SparseMatrix

Produces a SparseMatrix from its diagonal.

##### Parameters
 Vector d the diagonal entries
##### Returns
 SparseMatrix

Example:
• > $v = new Vector(1,2,3,4);>$D = diag($v);> print$D; (4) (0 1) (4) (1 2) (4) (2 3) (4) (3 4)
•
diag (m1, m2) → SparseMatrix

Returns a block diagonal matrix with blocks m1 and m2.

##### Parameters
 Matrix m1 Matrix m2
##### Returns
 SparseMatrix

Example:
• > $m1 = new Matrix([1,2],[3,4]);>$m2 = new Matrix([1,0,2],[3,4,0]);> $D = diag($m1,$m2);> print$D; (5) (0 1) (1 2) (5) (0 3) (1 4) 0 0 1 0 2 0 0 3 4 0
•
eigenvalues (A) → Vector<Float>

Eigenvalues of a matrix

##### Parameters
 Matrix A
##### Returns
 Vector
•
equal_bases (M1, M2) → Bool

Check whether both matrices are bases of the same linear subspace. Note: It is assumed that they are *bases* of the row space.

##### Parameters
 Matrix M1 Matrix M2
##### Returns
 Bool

Example:
• > $M1 = new Matrix([1,1,0],[1,0,1],[0,0,1]);>$M2 = new Matrix([1,0,0],[0,1,0],[0,0,1]);> print equal_bases($M1,$M2); true
•

Compute the Hadamard product of two matrices with same dimensions.

##### Parameters
 Matrix M1 Matrix M2
##### Returns
 Matrix
•
hermite_normal_form (M) → HermiteNormalForm

Computes the (column) Hermite normal form of an integer matrix. Pivot entries are positive, entries to the left of a pivot are non-negative and strictly smaller than the pivot.

##### Parameters
 Matrix M Matrix to be transformed.
##### Options
 Bool reduced If this is false, entries to the left of a pivot are left untouched. True by default
##### Returns
 HermiteNormalForm object H with H.hnf, H.companion and H.rank with M*H.companion=H.hnf and rank(M) = H.rank.

Example:
• The following stores the result for a small matrix M in H and then prints both hnf and companion:> $M = new Matrix<Integer>([1,2],[2,3]);>$H = hermite_normal_form($M);> print$H->hnf; 1 0 0 1> print $H->companion; -3 2 2 -1 • householder_trafo (b) → Matrix<Float> Householder transformation of Vector b. Only the orthogonal matrix reflection H is returned. ##### Parameters  Vector b ##### Returns  Matrix • inv (A) → Matrix Computes the inverse A-1 of an invertible matrix A using Gauss elimination. ##### Parameters  Matrix A ##### Returns  Matrix Example: • We save the inverse of a small matrix M in the variable$iM:> $M = new Matrix([1,2],[3,4]);>$iM = inv($M); To print the result, type this:> print$iM; -2 1 3/2 -1/2 As we can see, that is in fact the inverse of M.> print $M *$iM; 1 0 0 1
•
lineality_space (A) → Matrix

Compute the lineality space of a matrix A.

##### Parameters
 Matrix A
##### Returns
 Matrix

Example:
• > $M = new Matrix([1,1,0,0],[1,0,1,0]);> print lineality_space($M); 0 0 0 1
•
lin_solve (A, b) → Vector

Computes the vector x that solves the system Ax = b

##### Parameters
 Matrix A must be invertible Vector b
##### Returns
 Vector

Example:
• from the Wikipedia:> $A = new Matrix([3,2,-1],[2,-2,4],[-1,1/2,-1]);>$b = new Vector(1,-2,0);> print lin_solve($A,$b); 1 -2 -2
•
moore_penrose_inverse (M) → Matrix<Float>

Moore-Penrose Inverse of a Matrix

##### Parameters
 Matrix M
##### Returns
 Matrix
•
normalized (A) → Matrix<Float>

Normalize a matrix by dividing each row by its length (l2-norm).

##### Parameters
 Matrix A
##### Returns
 Matrix

Example:
• > $A = new Matrix<Float>([1.5,2],[2.5,2.5]);> print normalized($A); 0.6 0.8 0.7071067812 0.7071067812
•
null_space (A) → Matrix

Compute the null space of a matrix A.

##### Parameters
 Matrix A
##### Returns
 Matrix

Example:
• > $A = new Matrix([1,2,0],[2,0,2]);> print null_space($A); -1 1/2 1
•
null_space (b) → Matrix

Compute the null space of a vector b.

##### Parameters
 Vector b
##### Returns
 Matrix

Example:
• > $b = new Vector(1,2,3);> print null_space($b); -2 1 0 -3 0 1
•
null_space_integer (A) → SparseMatrix

Computes the lattice null space of the integer matrix A.

##### Parameters
 Matrix A
##### Returns
 SparseMatrix Has a lattice basis of the null space as rows.
•
ones_matrix <Element> (m, n) → Matrix<Element>

Creates a matrix with all elements equal to 1.

##### Type Parameters
 Element default: Rational.
##### Parameters
 Int m number of rows Int n number of columns
##### Returns
 Matrix

Example:
• The following creates an all-ones matrix with Rational coefficients.> $M = ones_matrix<Rational>(2,3);> print$M; 1 1 1 1 1 1
•
ones_vector <Element> (d) → Vector<Element>

Creates a vector with all elements equal to 1.

##### Type Parameters
 Element default: Rational.
##### Parameters
 Int d vector dimension. If omitted, a vector of dimension 0 is created, which can adjust itself when involved in a block matrix operation.
##### Returns
 Vector

Example:
• To create the all-ones Int vector of dimension 3, do this:> $v = ones_vector<Int>(3); You can print the result using the print statement:> print$v; 1 1 1
•
pluecker (V) → Vector

Compute the vector of maximal minors of a matrix. WARNING: interpretation different in tropical::lifted_pluecker

##### Parameters
 Matrix V
##### Returns
 Vector
•
project_to_orthogonal_complement (points, orthogonal)

Projects points into the orthogonal complement of a subspace given via an orthogonal basis. The given points will be overwitten.

##### Parameters
 Matrix points will be changed to orthogonal ones Matrix orthogonal basis of the subspace
•
qr_decomp (M) → Pair<Matrix,Matrix>

QR decomposition of a Matrix M with rows > cols

##### Parameters
 Matrix M
##### Returns
 Pair

Example:
• > $M = new Matrix<Float>([23,4],[6,42]);>$qr = qr_decomp($M);> print$qr->first; 0.9676172724 0.2524218971 0.2524218971 -0.9676172724> print $qr->second; 23.76972865 14.47218877 0 -39.63023785> print$qr->first * $qr->second ; 23 4 6 42 • rank (A) → Int Computes the rank of a matrix. ##### Parameters  Matrix A ##### Returns  Int • reduce (A, b) → Vector Reduce a vector with a given matrix using Gauss elimination. ##### Parameters  Matrix A Vector b ##### Returns  Vector • remove_zero_rows (m) → Matrix Remove all zero rows from a matrix. ##### Parameters  Matrix m ##### Returns  Matrix • singular_value_decomposition (M) → SingularValueDecomposition SVD decomposition of a Matrix. Computes the SVD of a matrix into a diagonal Marix (S), orthogonal square Matrix (U), orthogonal square Matrix (V), such that U*S*V^T=M The first element of the output array is S, the second U and the thrid V. ##### Parameters  Matrix M ##### Returns  SingularValueDecomposition Example: • >$M = new Matrix<Float>([1,2],[23,24]);> $SVD = singular_value_decomposition($M); The following prints the three matrices, seperated by newline characters.> print $SVD->left_companion ,"\n",$SVD->sigma ,"\n", $SVD->right_companion; 0.06414638608 0.9979404998 0.9979404998 -0.06414638608  33.31011547 0 0 0.6604600341  0.6909846321 -0.7228694476 0.7228694476 0.6909846321 • smith_normal_form (M, inv) → SmithNormalForm<Integer> Compute the Smith normal form of a given matrix M. M = LSR in normal case, or S = LMR in inverted case. ##### Parameters  Matrix M must be of integer type Bool inv optional, if true, compute the inverse of the companion matrices ##### Returns  SmithNormalForm Example: • >$M = new Matrix<Integer>([1,2],[23,24]);> $SNF = smith_normal_form($M); The following line prints the three matrices seperated by newline characters.> print $SNF->left_companion ,"\n",$SNF->form ,"\n", $SNF->right_companion; 1 0 23 1  1 0 0 -22  1 2 0 1 • solve_left (A, B) → Matrix Computes a matrix X that solves the system XA = B. This is useful, for instance, for computing the coordinates of some vectors with respect to a basis. The rows of the matrix solve_left(B,V) are the coordinates of the rows of V with respect to the rows of B. ##### Parameters  Matrix A Matrix B ##### Returns  Matrix Example: • Define the matrices>$V = new Matrix([[-4,2,2],[3,-2,-1]]);> $B = new Matrix([[-1,1,0],[0,-1,1]]); so that the rows of B are a basis of the subspace of vectors with zero coordinate sum. Then the rows of> print solve_left($B, $V); 4 2 -3 -1 contain the coordinates of the rows of V with respect to the rows of B. • solve_right (A, B) → Matrix Computes a matrix X that solves the system AX = B ##### Parameters  Matrix A Matrix B ##### Returns  Matrix Examples: • A non-degenerate example:>$A = new Matrix([[1,0,0],[1,1,0],[1,0,1],[1,1,1]]);> $B = new Matrix([[1,0,0],[1,0,1],[1,1,0],[1,1,1]]);> print solve_right($A,$B); 1 0 0 0 0 1 0 1 0 • A degenerate example:>$A = new Matrix([[1,0,0,0,0],[0,1,0,0,0],[1,0,1,0,0],[0,1,1,0,0]]);> $B = new Matrix([[0,1,0,0,0],[1,0,0,0,0],[0,1,1,0,0],[1,0,1,0,0]]);> print solve_right($A,$B); 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 • sqr (v) → Scalar Return the sum of the squared entries of a vector v. ##### Parameters  Vector v ##### Returns  Scalar • totally_unimodular (A) → Bool The matrix A is totally unimodular if the determinant of each square submatrix equals 0, 1, or -1. This is the naive test (exponential in the size of the matrix). For a better implementation try Matthias Walter's polymake extension at https://github.com/xammy/unimodularity-test/wiki/Polymake-Extension. ##### Parameters  Matrix A ##### Returns  Bool Example: • >$M = new Matrix<Int>([-1,-1,0,0,0,1],[1,0,-1,-1,0,0],[0,1,1,0,-1,0],[0,0,0,1,1,-1]);> print totally_unimodular($M); true • trace (A) → Int Computes the trace of a matrix. ##### Parameters  Matrix A ##### Returns  Int trace(A) Example: • >$M = new Matrix([1,2,3],[23,24,25],[0,0,1]);> print trace($M); 26 • transpose (A) → IncidenceMatrix Computes the transpose AT of an incidence matrix A, i.e., (aT)ij = aji. ##### Parameters  IncidenceMatrix A ##### Returns  IncidenceMatrix • transpose (A) → Matrix Computes the transpose AT of a matrix A, i.e., (aT)ij = aji. ##### Parameters  Matrix A ##### Returns  Matrix Example: • >$M = new Matrix([1,2,23],[23,22,21]);> $Mt = transpose($M);> print $Mt; 1 23 2 22 23 21 • unit_matrix <Element> (d) → SparseMatrix<Element> Creates a unit matrix of given dimension ##### Type Parameters  Element default: Rational ##### Parameters  Int d dimension of the matrix ##### Returns  SparseMatrix Examples: • The following stores the 3-dimensional unit matrix (ones on the diagonal and zeros otherwise) in a variable and prints it:>$M = unit_matrix(3);> print $M; (3) (0 1) (3) (1 1) (3) (2 1) • The following stores the 3-dimensional unit matrix (ones on the diagonal and zeros otherwise) from type Int in a variable and prints it:>$M = unit_matrix<Int>(3);> print $M->type->full_name; SparseMatrix<Int, Symmetric> • unit_vector <Element> (d, pos) → SparseVector<Element> Creates a SparseVector of given length d with a one entry at position pos and zeroes elsewhere. ##### Type Parameters  Element default: Rational ##### Parameters  Int d the dimension of the vector Int pos the position of the 1 ##### Returns  SparseVector # @example The following stores a vector of dimension 5 with a single 1 (as a Rational) at position 2: >$v = unit_vector(5,2); > print $v; | (5) (2 1) Examples: • The following stores a vector of dimension 5 with a single 1 (as a Int) at position 2:>$v = unit_vector<Int>(5,2);> print $v->type->full_name; SparseVector<Int> • The following concatenates a unit vector of dimension 3 with a 1 at position 2 and a unit vector of dimension 2 with a 1 at position 1:>$v = unit_vector(3,2) | unit_vector(2,1);> print $v; (5) (2 1) (4 1) • zero_matrix <Element> (i, j) → SparseMatrix<Element> Creates a zero matrix of given dimensions ##### Type Parameters  Element default: Rational ##### Parameters  Int i number of rows Int j number of columns ##### Returns  SparseMatrix Examples: • The following stores a 2x3 matrix with 0 as entries (from type Rational) in a variable and prints it:>$M = zero_matrix(2,3);> print $M; 0 0 0 0 0 0 • The following stores a 2x3 matrix with 0 as entries from type Int in a variable and prints its type:>$M = zero_matrix<Int>(2,3);> print $M->type->full_name; Matrix<Int, NonSymmetric> • zero_vector <Element> (d) → Vector<Element> Creates a vector with all elements equal to zero. ##### Type Parameters  Element default: Rational ##### Parameters  Int d vector dimension. If omitted, a vector of dimension 0 is created, which can adjust itself when involved in a block matrix operation. ##### Returns  Vector Examples: • The following stores a vector of dimension 5 with 0 as entries (from type Rational) in a variable and prints it:>$v = zero_vector(5);> print $v; 0 0 0 0 0 • The following stores a vector of dimension 5 with 0 as entries from type Int in a variable and prints its type:>$v = zero_vector<Int>(5);> print $v->type->full_name; Vector<Int> • The following concatenates a vector of dimension 2 of ones and a vector of length 2 of zeros:>$v = ones_vector(2) | zero_vector(2);> print $v; 1 1 0 0 • ### Set Operations This category contains functions performing operations on Sets. • incl (s1, s2) → Int Analyze the inclusion relation of two sets. ##### Parameters  Set s1 Set s2 ##### Returns  Int 0 if s1 = s2, -1 if s1 ⊂ s2, 1 if s1 ⊃ s2, 2 otherwise. Example: • >$s1 = new Set(1,2,3);> $s2 =$s1 - 1;> print incl($s1,$s2); 1> print incl($s2,$s1); -1> print incl($s1,$s1); 0> print incl($s2,$s1-$s2); 2 • range (a, b) → Set<Int> Create the Set {a, a+1, ..., b-1, b} for ab. See also: sequence ##### Parameters  Int a minimal element of the set Int b maximal element of the set ##### Returns  Set Example: • > print range(23,27); {23 24 25 26 27} • range_from (a) → Set<Int> Create an index range starting at a, the last index is implied by the context. To be used with minor, slice, and similar methods selecting a subset of elements. ##### Parameters  Int a start index ##### Returns  Set Example: • >$v=new Vector<Int>(10,20,30,40);> print $v->slice(range_from(2)); 30 40 • scalar2set (s) → Set<SCALAR> Returns the singleton set {s}. ##### Parameters  SCALAR s ##### Returns  Set Example: • > print scalar2set(23); {23} • select_subset (s, indices) → Set Returns the subset of s given by the indices. ##### Parameters  Set s Set indices ##### Returns  Set Example: • >$s = new Set<Int>(23,42,666,789);> $ind = new Set<Int>(0,2);>$su = select_subset($s,$ind);> print $su; {23 666} • sequence (a, c) → Set<Int> Create the Set {a, a+1, ..., a+c-1}. See also: range ##### Parameters  Int a the smallest element Int c the cardinality ##### Returns  Set Example: • > print sequence(23,6); {23 24 25 26 27 28} • ### Utilities Miscellaneous functions. • average (array) Returns the average value of the array elements. ##### Parameters  ARRAY array Example: • > print average([1,2,3]); 2 • bounding_box (m) → Matrix Compute a column-wise bounding box for the given Matrix m. ##### Parameters  Matrix m ##### Returns  Matrix a Matrix with two rows and m->cols columns; row(0) contains lower bounds, row(1) contains upper bounds. • distance_matrix (S, d) → Matrix Given a metric, return a triangle matrix whose (i,j)-entry contains the distance between point i and j of the point set S for i<j. All entrys below the diagonal are zero. The metric is passed as a perl subroutine mapping two input vectors to a real value. ##### Parameters  Matrix S CODE d ##### Returns  Matrix Example: • The following defines the perl subroutine dist as the euclidean metric and then saves the distance matrix of the 3-cubes vertices in the variable$M:> sub dist() {> my $v =$_[0] - $_[1];> return sqrt(new Float($v*$v)); }>$M = distance_matrix(polytope::cube(3)->VERTICES, \&dist);
•
fibonacci (m) → ARRAY

Returns the first m Fibonacci numbers.

##### Parameters
 Int m
##### Returns
 ARRAY
•
histogram (data) → Map<Element, Int>

Produce a histogram of an array: each different element value is mapped on the number of its occurences.

##### Parameters
 ARRAY data
##### Returns
 Map

Example:
• > $H = histogram([1,1,2,2,2,3,3,2,3,3,1,1,1,3,2,3]);> print$H; {(1 5) (2 5) (3 6)}
•
index_of (array) → HashMap<Array<Set<Int>>, Int>

Return a map indexing an array of sets

##### Parameters
 Array> array
##### Returns
 HashMap>, Int>

Example:
• > $s1 = new Set(1,2,3);>$s2 = $s1 - 1;>$a = new Array<Set>($s1,$s2,$s1);> print index_of($a); {({1 2 3} 2) ({2 3} 1)}
•
is_nonnegative (data) → Bool

Checks if a given sequence is nonnegative

##### Parameters
 ARRAY data
##### Returns
 Bool

Example:
• > print is_nonnegative([1,0,3]); 1
•
is_unimodal (data) → Bool

Checks if a given sequence is unimodal

##### Parameters
 ARRAY data (or Array)
##### Returns
 Bool

Example:
• > print is_unimodal([1,1,2,3,3,2,2,1]); 1 > print is_unimodal([3,3,2,-1]); 1> print is_unimodal([1,3,2,3]); 0
•
maximum (array)

Returns the maximal element of an array.

##### Parameters
 ARRAY array

Example:
• > print maximum([1,2,3,4,5,6,7,23]); 23
•
median (array)

Returns the median value of the array elements.

##### Parameters
 ARRAY array

Example:
• > print median([1,2,3,9]); 2.5
•
minimum (array)

Returns the minimal element of an array.

##### Parameters
 ARRAY array

Example:
• > print minimum([23,42,666]); 23
•
perturb_matrix (M, eps, not_hom) → Matrix

Perturb a given matrix M by adding a random matrix. The random matrix consists of vectors that are uniformly distributed on the unit sphere. Optionally, the random matrix can be scaled by a factor eps.

##### Parameters
 Matrix M Float eps the factor by which the random matrix is multiplied default value: 1 Bool not_hom if set to 1, the first column will also be perturbed; otherwise the first columns of the input matrix M and the perturbed one coincide (useful for working with homogenized coordinates); default value: 0 (homogen. coords)
##### Options
 Int seed controls the outcome of the random number generator; fixing a seed number guarantees the same outcome.
##### Returns
 Matrix
•
rand_perm (n) → Array<Int>

gives a random permutation

##### Parameters
 Int n
##### Options
 Int Seed
##### Returns
 Array random permutation
•

### Visualization

These functions are for visualization.

•
compose (vis_obj ...) → Visual::Container

Create a composite drawing of several objects.

##### Parameters
 Visual::Object vis_obj ... objects to be drawn together
##### Options
 String Title name of the whole drawing; per default the name of the first Object is taken. option list: Visual::Polygons::decorations
##### Returns
 Visual::Container if called in void context, immediately starts the preferred rendering program.

Example:
• Draw a pretty 8-pointed star:> compose(cube(2)->VISUAL, cross(2,sqrt(2))->VISUAL, Title=>"A pretty star.", VertexLabels=>"hidden");
•
compose (vis_container, vis_obj ...) → Visual::Container

Add new objects to a composite drawing.

##### Parameters
 Visual::Container vis_container drawing produced by some visualization function Visual::Object vis_obj ... objects to be added
##### Options
 String Title new name for the drawing any decorations to be applied to all components as default values.
##### Returns
 Visual::Container if called in void context, immediately starts the preferred rendering program.
•
geomview (vis_obj ...)

Run geomview to display given visual objects.

##### Parameters
 Visual::Object vis_obj ... objects to display
##### Options
 String File "filename" or "AUTO" Store the objects in a gcl (geomview control language) file instead of starting the interactive GUI. The geometric data in OFF format is embedded in the Lisp-style commands, but can be easily extracted using any text editor, if needed. The .gcl suffix is automatically added to the file name. Specify AUTO if you want the filename be automatically derived from the drawing title. You can also use any expression allowed for the open function, including "-" for terminal output, "&HANDLE" for an already opened file handle, or "| program" for a pipe.
•
javaview (vis_obj ...)

Run JavaView with the given visual objects.

Contained in extension javaview.
##### Parameters
 Visual::Object vis_obj ... objects to display
##### Options
 String File "filename" or "AUTO" Store the object description in a JVX file without starting the interactive GUI. The .jvx suffix is automatically added to the file name. Specify AUTO if you want the filename be automatically derived from the drawing title. You can also use any expression allowed for the open function, including "-" for terminal output, "&HANDLE" for an already opened file handle, or "| program" for a pipe.
•
jreality (vis_obj ...)

Run jReality to display given visual objects.

Contained in extension jreality.
##### Parameters
 Visual::Object vis_obj ... objects to display
##### Options
 String File "filename" or "AUTO" Store the object description in a bean shell source file without starting the interactive GUI. The .bsh suffix is automatically added to the file name. Specify AUTO if you want the filename be automatically derived from the drawing title. You can also use any expression allowed for the open function, including "-" for terminal output, "&HANDLE" for an already opened file handle, or "| program" for a pipe.
•
postscript (vis_obj ...)

Create a Postscript (tm) drawing with the given visual objects.

##### Parameters
 Visual::Object vis_obj ... objects to draw
##### Options
 String File "filename" or "AUTO" Store the drawing in a file without starting the viewer. The .ps suffix is automatically added to the file name. Specify AUTO if you want the filename be automatically derived from the drawing title. You can also use any expression allowed for the open function, including "-" for terminal output, "&HANDLE" for an already opened file handle, or "| program" for a pipe.
•
povray (vis_obj ...)

Run POVRAY to display given visual objects.

##### Parameters
 Visual::Object vis_obj ... objects to display
##### Options
 String File "filename" or "AUTO" Store the object description in a POVRAY source file without actual rendering. The .pov suffix is automatically added to the file name. Specify AUTO if you want the filename be automatically derived from the drawing title. You can also use any expression allowed for the open function, including "-" for terminal output, "&HANDLE" for an already opened file handle, or "| program" for a pipe.
•
sketch (vis_obj ...)

Produce a Sketch input file with given visual objects.

##### Parameters
 Visual::Object vis_obj ... objects to display
##### Options
 String File "filename" or "AUTO" For the file name you can use any expression allowed for the open function, including "-" for terminal output, "&HANDLE" for an already opened file handle, or "| program" for a pipe. Real file names are automatically completed with the .sk suffix if needed. An automatically generated file name is displayed in the verbose mode.
•
static (vis_obj) → Visual::Object

Suppress creation of dynamic (interactive) scenes.

##### Parameters
 Visual::Object vis_obj drawing, e.g. created by VISUAL_GRAPH or SCHLEGEL.
##### Returns
 Visual::Object if called in void context, immediately starts the preferred rendering program.
•
svg (vis_obj ...)

Create a Svg drawing with the given visual objects.

##### Parameters
 Visual::Object vis_obj ... objects to draw
##### Options
 String File "filename" or "AUTO" Store the drawing in a file without starting the viewer. The .svg suffix is automatically added to the file name. Specify AUTO if you want the filename be automatically derived from the drawing title. You can also use any expression allowed for the open function, including "-" for terminal output, "&HANDLE" for an already opened file handle, or "| program" for a pipe.
•
threejs (vis_obj)

Produce an html file with given visual objects.

##### Parameters
 Visual::Object vis_obj object to display
##### Options
 String File "filename" or "AUTO" For the file name you can use any expression allowed for the open function, including "-" for terminal output, "&HANDLE" for an already opened file handle, or "| program" for a pipe. Real file names are automatically completed with the .html suffix if needed. An automatically generated file name is displayed in the verbose mode.
•
tikz (vis_obj)

Produce a TikZ file with given visual objects.

##### Parameters
 Visual::Object vis_obj object to display
##### Options
 String File "filename" or "AUTO" For the file name you can use any expression allowed for the open function, including "-" for terminal output, "&HANDLE" for an already opened file handle, or "| program" for a pipe. Real file names are automatically completed with the .tikz suffix if needed. An automatically generated file name is displayed in the verbose mode.
•
x3d (vis_obj ...)

Create an X3D drawing with the given visual objects.

##### Parameters
 Visual::Object vis_obj ... objects to draw
##### Options
 String File "filename" or "AUTO" Store the drawing in a file without starting the viewer. The .x3d suffix is automatically added to the file name. Specify AUTO if you want the filename be automatically derived from the drawing title. You can also use any expression allowed for the open function, including "-" for terminal output, "&HANDLE" for an already opened file handle, or "| program" for a pipe.

## Property Types

•

### Algebraic Types

This category contains all "algebraic" types, such as matrices, vectors, polynomials, rings, ...

•
all_rows_or_cols

Use the keyword "All" for all rows or columns, e.g. when constructing a minor.

•
Matrix <Element, Sym>
UNDOCUMENTED
##### Type Parameters
 Element default: Rational Sym default: NonSymmetric

#### User Methods of Matrix

•
anti_diagonal (i) → Vector<Element>

Returns the anti-diagonal of the matrix.

##### Parameters
 Int i i=0: the main anti_diagonal (optional) i>0: the i-th anti_diagonal below the main anti_diagonal i<0: the i-th anti_diagonal above the main anti_diagonal
##### Returns
 Vector
•
clear (r, c)

Change the dimensions setting all elements to 0.

##### Parameters
 Int r new number of rows Int c new number of columns
•
col (i) → Vector<Element>

Returns the i-th column.

##### Parameters
 Int i
##### Returns
 Vector
•
cols () → Int

Returns the number of columns.

##### Returns
 Int
•
diagonal (i) → Vector<Element>

Returns the diagonal of the matrix.

##### Parameters
 Int i i=0: the main diagonal (optional) i>0: the i-th diagonal below the main diagonal i<0: the i-th diagonal above the main diagonal
##### Returns
 Vector
•
div_exact (a) → Matrix

Divides every entry by a (assuming that every entry is divisible by a).

##### Parameters
 Int a
##### Returns
 Matrix
•
elem (r, c) → Element

Returns an element of the matrix. The return value is an lvalue', that is, it can be modified if the matrix object is mutable.

##### Parameters
 Int r the row index Int c the column index
##### Returns
 Element
•
minor (r, c) → Matrix

Returns a minor of the matrix containing the rows in r and the columns in c. You can pass All if you want all rows or columns and ~ for the complement of a set. E.g.

$A->minor(All, ~[0]); will give you the minor of a matrix containing all rows and all but the 0-th column. ##### Parameters  Set r the rows Set c the columns ##### Returns  Matrix • resize (r, c) Change the dimensions; when growing, set added elements to 0. ##### Parameters  Int r new number of rows Int c new number of columns • row (i) → Vector<Element> Returns the i-th row. ##### Parameters  Int i ##### Returns  Vector • rows () → Int Returns the number of rows. ##### Returns  Int • Plucker <Scalar> UNDOCUMENTED ##### Type Parameters  Scalar default: Rational #### User Methods of Plucker • Polynomial <Coefficient, Exponent> UNDOCUMENTED ##### Type Parameters  Coefficient default: Rational Exponent default: Int #### User Methods of Polynomial • coefficients_as_vector () → Vector<Coefficient> The vector of all coefficients. The sorting agrees with monomials_as_matrix. ##### Returns  Vector • constant_coefficient () → Int The constant coefficient. ##### Returns  Int • deg () → Int The degree of the polynomial ##### Returns  Int • embed (nvars) → Polynomial Embed the Polynomial in a polynomial ring with the given number of variables. The old variables will be put at the beginning, for more control use mapvars. ##### Parameters  Int nvars new number of variables ##### Returns  Polynomial Example: • >$pm = new Polynomial("1+x_0^2*x_3+x_1+3*x_3");> print $pm->project([1,3]); x_0 + 4*x_1 + 1 • get_var_names () set the variable names • initial_form (v) → Polynomial The initial form with respect to a weight-vector v ##### Parameters  Vector v weights ##### Returns  Polynomial • lc () → Int The leading coefficient. ##### Returns  Int • lm () → Int The exponent of the leading monomial. ##### Returns  Int • mapvars (indices, nvars) → Polynomial Map the variables of the Polynomial to the given indices. The same index may bei given multiple times and also some may be omitted for embedding with a higher number of variables. The length of the given Array must be the same as the number of variables of the original polynomial. ##### Parameters  Array indices indices of the target variables Int nvars new number of variables, default: maximal index ##### Returns  Polynomial Example: • >$pm = new Polynomial("1+x_0^2*x_3+x_1+3*x_3");> print $pm->mapvars([0,1,1,4],6); x_0^2*x_4 + x_1 + 3*x_4 + 1 • monomial (var_index, n_vars) construct a monomial of degree 1 with a given variable ##### Parameters  Int var_index index of the variable Int n_vars number of variables • monomials_as_matrix () → SparseMatrix<Exponent> The matrix of all exponent vectors (row-wise). The sorting agrees with coefficients_as_vector. ##### Returns  SparseMatrix • n_vars () → Int Number of variables ##### Returns  Int • print_ordered (m) Print a polynomial with terms sorted according to a given Matrix m. ##### Parameters  Matrix m • project () → Polynomial Project the Polynomial to the given list of variables. The number of variables will be reduced to the number of indices, i.e. the variables will be renamed. Keeping the names of the variables is possible by using substitute with a Map. ##### Returns  Polynomial Example: • >$pm = new Polynomial("1+x_0^2*x_3+x_1+3*x_3");> print $pm->project([1,3]); x_0 + 4*x_1 + 1 • reset_var_names () reset the variable names according to the default naming scheme • set_var_names () set the variable names • substitute () Substitute a list of values in a Polynomial with Int exponents. Either with an array of values, or with a Map mapping variable indices to values. Example: • >$pm = new Polynomial("1+x_0^2*x_3+x_1+3*x_3");> print $pm->substitute([0,11,2,3]); 21>$map = new Map<Int,Rational>([0,5/2],[1,1/5],[2,new Rational(7/3)]);> print $pm->substitute($map); 37/4*x_3 + 6/5
•
trivial () → Bool

The polynomial is zero.

##### Returns
 Bool
•
PuiseuxFraction <MinMax, Coefficient, Exponent>
UNDOCUMENTED
##### Type Parameters
 MinMax type of tropical addition: either Min or Max Coefficient default: Rational Exponent default: Rational

#### User Methods of PuiseuxFraction

•
evaluate (m, x, exp) → Matrix<Coefficient>

Evaluate all PuiseuxFractions in a Matrix at a Rational number (x^exp). Let explcm be the lcm of the denominators of all exponents. If there are no denominators or explcm divides exp, then the evaluation is computed exactly. Otherwise, some rational number close to the root (x^exp)^-explcm will be chosen via an intermediate floating point number.

##### Parameters
 Matrix m Coefficient x Int exp (default: 1)
##### Returns
 Matrix
•
evaluate (v, x, exp) → Vector<Coefficient>

Evaluate all PuiseuxFractions in a Vector at a Rational number (x^exp). Let explcm be the lcm of the denominators of all exponents. If there are no denominators or explcm divides exp, then the evaluation is computed exactly. Otherwise, some rational number close to the root (x^exp)^-explcm will be chosen via an intermediate floating point number.

##### Parameters
 Vector v Coefficient x Int exp (default: 1)
##### Returns
 Vector
•
evaluate (x, exp) → Coefficient

Evaluate a PuiseuxFraction at a Rational number (x^exp). Let explcm be the lcm of the denominators of all exponents. If there are no denominators or explcm divides exp, then the evaluation is computed exactly. Otherwise, some rational number close to the root (x^exp)^-explcm will be chosen via an intermediate floating point number.

##### Parameters
 Coefficient x Int exp (default: 1)
##### Returns
 Coefficient
•
evaluate_float (x) → Float

Approximate evaluation at x

##### Parameters
 Float x
##### Returns
 Float
•
evaluate_float (m, x) → Float

Approximate evaluation of a Matrix at x

##### Parameters
 Matrix m Float x
##### Returns
 Float
•
evaluate_float (v, x) → Float

Approximate evaluation of a Vector at x

##### Parameters
 Vector v Float x
##### Returns
 Float
•
val () → TropicalNumber<MinMax>

The valuation.

##### Returns
 TropicalNumber
•
RationalFunction <Coefficient, Exponent>

the same with type deduction

##### Type Parameters
 Coefficient default: Rational Exponent default: Int
•
SparseMatrix <Element, Sym>

A SparseMatrix is a two-dimensional associative array with row and column indices as keys; elements equal to the default value (ElementType(), which is 0 for most numerical types) are not stored, but implicitly encoded by the gaps in the key set. Each row and column is organized as an AVL-tree.

Use dense to convert this into its dense form.

You can create a new SparseMatrix by entering its entries row by row, as a list of SparseVectors e.g.:

$A = new SparseMatrix<Int>(<< '.'); (5) (1 1) (5) (4 2) (5) (5) (0 3) (1 -1) . derived from: Matrix ##### Type Parameters  Element Sym one of Symmetric or NonSymmetric, default: NonSymmetric #### User Methods of SparseMatrix • resize (r, c) Resize the matrix All elements in added rows and/or columns are implicit zeros. ##### Parameters  Int r new number of rows Int c new number of columns • squeeze () Removes empty rows and columns. The remaining rows and columns are renumbered without gaps. • squeeze_cols () Removes empty columns. The remaining columns are renumbered without gaps. • squeeze_rows () Removes empty rows. The remaining rows are renumbered without gaps. • SparseVector <Element> A SparseVector is an associative container with element indices (coordinates) as keys; elements equal to the default value (ElementType(), which is 0 for most numerical types) are not stored, but implicitly encoded by the gaps in the key set. It is based on an AVL tree. The printable representation of a SparseVector looks like a sequence (l) (p1 v1) ... (pk vk), where l is the dimension of the vector and each pair (pi vi) denotes an entry with value vi at position pi. All other entries are zero. Use dense to convert this into its dense form. You can create a new SparseVector by entering its printable encoding as described above, e.g.:$v = new SparseVector<Int>(<< '.');
(6) (1 1) (2 2)
.
derived from: Vector
##### Type Parameters
 Element

#### User Methods of SparseVector

•
size () → Int

The number of non-zero entries.

##### Returns
 Int
•
UniPolynomial <Coefficient, Exponent>

A class for univariate polynomials.

##### Type Parameters
 Coefficient default: Rational Exponent default: Int

#### User Methods of UniPolynomial

•
coefficients_as_vector () → Vector<Coefficient>

The vector of all coefficients. The sorting agrees with monomials_as_vector.

##### Returns
 Vector
•
constant_coefficient () → Int

The constant coefficient.

##### Returns
 Int
•
deg () → Exponent

The highest degree occuring in the polynomial.

##### Returns
 Exponent
•
evaluate (x, exp) → Coefficient

Evaluate a UniPolynomial at a number (x^exp). Note that for non-integral exponents this may require intermediate floating point computations depending on the input: Let explcm be the lcm of the denominators of all exponents. If there are no denominators or explcm divides exp, then the evaluation is computed exactly. Otherwise, some rational number close to the root (x^exp)^-explcm will be chosen via an intermediate floating point number.

##### Parameters
 Coefficient x Exponent exp (default: 1)
##### Returns
 Coefficient
•
evaluate_float (x) → Float

Approximate evaluation of the polynomial at a Float x.

##### Parameters
 Float x
##### Returns
 Float
•
get_var_names ()

get the variable name

•
lc () → Int

##### Returns
 Int
•
lower_deg () → Exponent

The lowest degree occuring in the polynomial.

##### Returns
 Exponent
•
monomial ()

create a monomial of degree 1

•
monomials_as_vector () → Vector<Exponent>

The vector of all exponents. The order agrees with coefficients_as_vector.

##### Returns
 Vector
•
n_vars ()

Number of variables

•
print_ordered (x)

Print a polynomial with terms sorted according to exponent*x.

##### Parameters
 Exponent x
•
reset_var_names ()

reset the variable name according to the default naming scheme

•
set_var_names ()

set the variable name

•
substitute ()

Substitute some value in a UniPolynomial with Int exponents. When all exponents are positive the argument can be any scalar, matrix or polynomial type. With negative exponents, polynomials are not supported (use RationalFunction instead) and any given matrix must be invertible

•
Vector <Element>

A type for vectors with entries of type Element.

You can perform algebraic operations such as addition or scalar multiplication.

You can create a new Vector by entering its elements, e.g.:

$v = new Vector<Int>(1,2,3); or$v = new Vector<Int>([1,2,3]);
##### Type Parameters
 Element

#### User Methods of Vector

•
dim () → Int

The length of the vector

##### Returns
 Int
•
div_exact (a) → Vector

Divides every entry by a (assuming that every entry is divisible by a).

##### Parameters
 Int a
##### Returns
 Vector
•
slice (s) → Vector

Returns a Vector containing all entries whose index is in a Set s.

##### Parameters
 Set s indices to select from the vector
##### Returns
 Vector
•

### Arithmetic

These types are needed as return types of arithmetic computations.

•
Div <Scalar>

The complete result of an integral division, quotient and remainder.

##### Type Parameters
 Scalar

#### User Methods of Div

•
quot () → Scalar

The quotient.

##### Returns
 Scalar
•
rem () → Scalar

The remainder.

##### Returns
 Scalar
•
ExtGCD

The complete result of the calculation of the greatest common divisor of two numbers a and b:

g=gcd(a,b)
g=a*p+b*q
a=g*k1
b=g*k2

#### User Methods of ExtGCD

•
g () → Int

The greatest common divisor of a and b.

##### Returns
 Int
•
k1 () → Int

The factor of a.

##### Returns
 Int
•
k2 () → Int

The factor of b.

##### Returns
 Int
•
p () → Int

The co-factor of a.

##### Returns
 Int
•
q () → Int

The co-factor of b.

##### Returns
 Int
•
Max

•
Min

#### User Methods of Min

•
UNDOCUMENTED

#### User Methods of TropicalNumber

•
orientation () → Int

The orientation of the associated addition, i.e. +1 if the corresponding 0 is +inf -1 if the corresponding 0 is -inf

##### Returns
 Int
•
zero () → Scalar

The zero element of the tropical semiring of this element.

##### Returns
 Scalar
•

### Artificial

These types are auxiliary artifacts helping to build other classes, primarily representing template parameters or enumeration constants. They should not be used alone as property types or function arguments. In the most cases they won't even have user-accessible constructors.

•

### Basic Types

This category contains all basic types, in particular those that wrap C++, GMP or perl types such as Int, Integer, Rational, Long, Float, Array, String, ...

•
AccurateFloat

A wrapper for AccurateFloat.

•
Array <Element>

An array with elements of type Element. Corresponds to the C++ type Array.

##### Type Parameters
 Element

#### User Methods of Array

•
size () → Int

Returns the size.

##### Returns
 Int
•
Bool

Corresponds to the C++ type bool.

•
Float

Corresponds to the C++ type double.

#### User Methods of Float

•
inf ()

produce an infinitely large positive value

•
minus_inf ()

produce an infinitely large negative value

•
Int

Corresponds to the C++ type int.

•
Integer

An integer of arbitrary size.

#### User Methods of Integer

•
inf ()

produce an infinitely large positive value

•
minus_inf ()

produce an infinitely large negative value

•
List <Element>

Corresponds to the C++ type std::list.

##### Type Parameters
 Element
•
Long

Corresponds to the C++ type long. This type is primarily needed for automatic generated function wrappers, because perl integral constants are always kept as a long.

derived from: Int
•
Pair <First, Second>

Corresponds to the C++ type std::pair.

##### Type Parameters
 First Second
•

You can construct the value a+b$$\sqrt r$$ via QuadraticExtension(a, b, r) (where a, b, r are of type Field).

##### Type Parameters
 Field default: Rational
•
Rational

A class for rational numbers. You can easily create a Rational like that: $x = 1/2; #### User Methods of Rational • inf () Produce an infinitely large positive value. • minus_inf () Produce an infinitely large negative value. • String Corresponds to the C++ type std::string. • Text Plain text without any imposed structure. • ### Database Here you can find the functions to access the polymake database. • DBCursor A database cursor. Initialize it with a database name, a collection name and a query. You can then iterate over the objects matching the query with next. It lazily fetches more objects from the database server. (Note that you have to create a new cursor if you want to start from the beginning.) • MongoClient A handler for the polyDB database internally controlling the connection to the MongoDB database • ### Graph Types This contains all property types that are related to graphs. • EdgeHashMap <Dir, Element> Sparse mapping of edges to data items. derived from: GraphMap ##### Type Parameters  Dir Element data associated with edges #### User Methods of EdgeHashMap • edge (from, to) Access the data associated with an edge between two given nodes. The new data element is created on demand. ##### Parameters  Int from source node Int to target node • erase (from, to) Delete the data associated with an edge between two given nodes. ##### Parameters  Int from source node Int to target node • find (from, to) → Iterator Access the data associated with an edge between two given nodes. ##### Parameters  Int from source node Int to target node ##### Returns  Iterator pointing to the data element (must be dereferenced as${...}) or undef if the element does not exist.
•
EdgeMap <Dir, Element>

Dense mapping of edges to data items.

derived from: GraphMap
##### Type Parameters
 Dir kind of the host graph, Undirected, Directed, UndirectedMulti, or DirectedMulti Element data associated with edges

#### User Methods of EdgeMap

•
edge (from, to)

Access the data associated with an edge between two given nodes.

##### Parameters
 Int from source node Int to target node
•
Graph <Dir>
UNDOCUMENTED
##### Type Parameters
 Dir one of Undirected, Directed, UndirectedMulti or DirectedMulti, default: Undirected

#### User Methods of Graph

•

In a multigraph, creates a new edge connecting two given nodes. In a normal graph, creates a new edge only if the nodes were not connected yet. Returns the index of the (new) edge.

##### Returns
 Int
•

Add a new node without incident edes.

##### Returns
 Int index of the new node
•

Returns the set of indices of nodes adjacent to node.

##### Parameters
 Int node
##### Returns
 Set
•

Returns an iterator visiting all (parallel) edges connecting two given nodes.

##### Returns
 Iterator
•
contract_edge (node1, node2)

Contract the edge(s) between node1 and node2. Reconnects all edges from node2 to node1, deleting the edge(s) between them and, finally, deleting node2.

##### Parameters
 Int node1 Int node2
•
degree (node) → Int

Returns the number of edges incident to node.

##### Parameters
 Int node
##### Returns
 Int
•

Deletes all edges in a multigraph connecting two given nodes.

##### Parameters
•

Deletes the edge connecting two given nodes, if there was one. In a multigraph, deletes one arbitrary edge from the parallel bundle.

##### Parameters
•
delete_edge (iterator)

Delete the edge in a multigraph pointed to by the given iterator

##### Parameters
 Iterator iterator as returned by all_edges.
•
delete_node (node)

Deletes all edges incident to the given node and marks it as invalid. The numeration of other nodes stays unchanged.

##### Parameters
 Int node
•
dim () → Int

Returns the maximal node index + 1. If the graph does not have gaps caused by node deletion, the result is equivalent to nodes().

##### Returns
 Int
•

Returns the index of the edge connecting two given nodes. The edge is created if was not there. In a multigraph, an arbitrary edge from the parallel bundle will be picked.

##### Returns
 Int
•
edges () → Int

Get the total number of edges.

##### Returns
 Int
•

Checks whether two given nodes are connected by (at least) one edge.

##### Returns
 Bool
•
has_gaps () → Bool

Returns true if some nodes have been deleted since the last squeeze operation.

##### Returns
 Bool
•
invalid_node (node) → Bool

Returns true if the given node index is either out of valid range or points to a formerly deleted node.

##### Parameters
 Int node
##### Returns
 Bool
•

Returns the set of indices of the nodes that have an edge heading to node.

##### Parameters
 Int node
##### Returns
 Set
•
in_degree (node) → Int

Returns the number of edges heading to node.

##### Parameters
 Int node
##### Returns
 Int
•
in_edges (node) → EdgeList

Returns a sequence of edges heading to (in Directed case) or incident to (in Undirected case) node.

##### Parameters
 Int node
##### Returns
 EdgeList
•
nodes () → Int

Get the total number of nodes.

##### Returns
 Int
•
node_exists (node) → Bool

Check whether the node with given index exists.

##### Parameters
 Int node
##### Returns
 Bool
•
node_out_of_range (node) → Bool

Returns true if the given node index is out of valid range.

##### Parameters
 Int node
##### Returns
 Bool
•

Returns the set of indices of the nodes with an edge arriving from node.

##### Parameters
 Int node
##### Returns
 Set
•
out_degree (node) → Int

Returns the number of edges leaving node.

##### Parameters
 Int node
##### Returns
 Int
•
out_edges (node) → EdgeList

Returns a sequence of edges leaving (in Directed case) or incident to (in Undirected case) node.

##### Parameters
 Int node
##### Returns
 EdgeList
•
permute_inv_nodes ()

permute the nodes param perm inverse permutation of node indexes

•
permute_nodes ()

permute the nodes param perm permutation of node indexes

•
squeeze ()

Renumbers the valid nodes as to eliminate all gaps left after deleting.

•
squeeze_isolated ()

Deletes all nodes of degree 0, then renumbers the remaining nodes without gaps.

•
GraphMap <Dir, Element>

The common abstract base class for all kinds of associative containers that can be attached to a Graph.

##### Type Parameters
 Dir kind of the host graph: Undirected, Directed, UndirectedMulti, or DirectedMulti Element data associated with nodes or edges
•
NodeHashMap <Dir, Element>

Sparse mapping of nodes to data items.

derived from: GraphMap
##### Type Parameters
 Dir Element data associated with nodes
•
NodeMap <Dir, Element>

Dense mapping of nodes to data items.

derived from: GraphMap
##### Type Parameters
 Dir kind of the host graph, Undirected, Directed, UndirectedMulti, or DirectedMulti Element data associated with nodes
•

### Linear Algebra

These types are needed as return types of algebraic computations.

•
HermiteNormalForm

Complete result of the Hermite normal form computation. Contains the following fields: Matrix<Scalar> hnf: the Hermite normal form N of M SparseMatrix<Scalar> companion: unimodular Matrix R such that M*R = N. Int rank: rank of M

•
SingularValueDecomposition

Complete result of the singular value decomposition of a matrix M, such that left_companion * sigma * transpose(right_companion) = M Contains the following fields:

Matrix<Float> sigma: the diagonalized matrix
Matrix<Float> left_companion: matrix of left singular vectors
Matrix<Float> right_companion: matrix of right singular vectors
•
SmithNormalForm

Complete result of the Smith normal form computation. Contains the following fields:

SparseMatrix<Scalar> form: the Smith normal form S of the given matrix M
List<Pair<Scalar, Int>> torsion: absolute values of the entries greater than 1 of the diagonal together with their multiplicity
Int rank: rank of M
SparseMatrix<Scalar> left_companion, right_companion: unimodular matrices L and R such that

M = LSR in normal case, or S = LMR in inverted case (as specified in the call to smith_normal_form function).

•

### Set Types

In this category you find all property types related to sets, such as Set, Map, HashMap, IncidenceMatrix, ...

•
ApproximateSet <Element>

A specialization of Sets containing elements of type Element, but where equality is enforced only up to a global epsilon.

You can for example create a new ApproximateSet with two elements by:

$s = new ApproximateSet(1.1, 1.2, 1.200000001); derived from: Set ##### Type Parameters  Element default: Float • Bitset Container class for dense sets of integers. A special class optimized for representation of a constrained range of non-negative integer numbers. derived from: Set<Int> • FacetList A FacetList is a collection of sets of integral numbers from a closed contiguous range [0..n-1]. The contained sets usually encode facets of a simplicial complex, with set elements corresponding to vertices of a complex, therefore the name. From the structural perspective, FacetList is interchangeable with IncidenceMatrix, but they significantly differ in supported operations and their performance. IncidenceMatrix offers fast random access to elements, while FacetList is optimized for finding, inserting, and deleting facets fulfilling certain conditions like all subsets or supersets of a given vertex set. On perl side, FacetList behaves like a sequence of Set<Int> without random access to facets. Facets are visited in chronological order. Each facet has a unique integral ID generated at the moment of insertion. The IDs can be obtained via call to index() of iterators created by find() methods. #### User Methods of FacetList • erase (f) → Bool Remove a facet. ##### Parameters  Set f facet to remove ##### Returns  Bool whether a facet existed before • eraseSubsets (s) → Int Remove all subsets of a given set ##### Parameters  Set s filter for removal ##### Returns  Int number of removed facets • eraseSupersets (s) → Int Remove all supersets of a given set ##### Parameters  Set s filter for removal ##### Returns  Int number of removed facets • find (f) → Iterator Look up a facet. ##### Parameters  Set f facet to find ##### Returns  Iterator pointing to the facet or in an invalid state • findSubsets (s) → Iterator Find all subsets of a given set. ##### Parameters  Set s ##### Returns  Iterator all facets equal to or included in s, visited in lexicographical order • findSupersets (s) → Iterator Find all supersets of a given set. ##### Parameters  Set s ##### Returns  Iterator all facets equal to or including s, visited in reverse chronological order • insert (f) Add a new facet. It may be a proper subset or a proper superset of existing facets. It must not be empty or coincide with any existing facet. ##### Parameters  Set f facet to add. • insertMax (f) → Bool Add a new facet if and only if there are no facets including it. If this holds, remove all facets that are included in the new one. ##### Parameters  Set f facet to add ##### Returns  Bool whether the new facet was really included. • insertMin (f) → Bool Add a new facet if and only if there are no facets included in it. If this holds, remove all facets including the new facet. ##### Parameters  Set f facet to add ##### Returns  Bool whether the new facet was really included. • n_vertices () → Int The number of vertices ##### Returns  Int • size () → Int The number of facets in the list. ##### Returns  Int • HashMap <Key, Value> An unordered map based on a hash table. Its interface is similar to that of Map, but the order of elements is not stable across polymake sessions and does not correlate with key values. Accessing and inserting a value by its key works in constant time O(1). You can create a new HashMap mapping Ints to Strings by$myhashmap = new HashMap<Int, String>([1, "Monday"], [2, "Tuesday"]);

On the perl side HashMaps are treated like hashrefs. You can work with a HashMap like you work with a Map (keeping in mind differences in performance and memory demand).

##### Type Parameters
 Key type of the key values Value type of the mapped value

#### User Methods of HashMap

•
size () → Int

Size of the map

##### Returns
 Int
•
HashSet <Element>

An unordered set of elements based on a hash table. Can be traversed as a normal container, but the order of elements is not stable across polymake sessions, even if the set is restored from the same data file every time.

##### Type Parameters
 Element

#### User Methods of HashSet

•
size () → Int

The cardinality of the set.

##### Returns
 Int
•
IncidenceMatrix <Sym>

A 0/1 incidence matrix.

##### Type Parameters
 Sym one of Symmetric or NonSymmetric, default: NonSymmetric

#### User Methods of IncidenceMatrix

•
col (i) → SparseVector<Int>

Returns the i-th column.

##### Parameters
 Int i
##### Returns
 SparseVector
•
cols () → Int

Returns the number of columns.

##### Returns
 Int
•
elem (r, c) → Bool

Returns an element of the matrix as a boolean value. The return value is an `lvalue', that is, it can be assigned to, flipped, etc. if the matrix object is mutable.

##### Parameters
 Int r the row index Int c the column index
##### Returns
 Bool
•
minor (r, c) → IncidenceMatrix

Returns a minor of the matrix containing the rows in r and the columns in c. You can pass All if you want all rows or columns and ~ for the complement of a set. For example,

$A->minor(All, ~[0]); will give you the minor of a matrix containing all rows and all but the 0-th column. ##### Parameters  Set r the rows Set c the columns ##### Returns  IncidenceMatrix • row (i) → SparseVector<Int> Returns the i-th row. ##### Parameters  Int i ##### Returns  SparseVector • rows () → Int Returns the number of rows. ##### Returns  Int • squeeze () Removes empty rows and columns. The remaining rows and columns are renumbered without gaps. • squeeze_cols () Removes empty columns. The remaining columns are renumbered without gaps. • squeeze_rows () Removes empty rows. The remaining rows are renumbered without gaps. • Map <Key, Value> Maps are sorted associative containers that contain unique key/value pairs. Maps are sorted by their keys. Accessing or inserting a value needs logarithmic time O(log n), where n is the size of the map. You can create a new Map mapping Ints to Strings by$mymap = new Map<Int, String>([1, "Monday"], [2, "Tuesday"]);

On the perl side Maps are treated like hashrefs. You can add a new key/value pair by

$mymap->{3} = "Wednesday"; (If the key is already contained in the Map, the corresponding value is replaced by the new one.) or ask for the value of a key by print$mymap->{1};
##### Type Parameters
 Key type of the key values Value type of the mapped value
•
PowerSet <Element>

A Set whose elements are of type Set<Element>.

derived from: Set
##### Type Parameters
 Element default: Int
•
Set <Element>

A type for ordered sets containing elements of type Element.

You can for example create a new Set by:

$s = new Set(2, 3, 5, 7); You can perform set theoretic operations:$s1 + $s2 union$s1 * $s2 intersection$s1 - $s2 difference$s1 ^ \$s2 symmetric difference
##### Type Parameters
 Element default: Int

#### User Methods of Set

•
back () → Int

The last element of the set, that is, the largest element

##### Returns
 Int
•
collect (e) → Bool

Add to the set, report true if existed formerly.

##### Parameters
 Element e element to insert into the set
##### Returns
 Bool
•
contains (e) → Bool

Check if e is contained in the set.

##### Parameters
 Element e element check for
##### Returns
 Bool
•
front () → Int

The first element of the set, that is, the smallest element

##### Returns
 Int
•
size () → Int

The cardinality of the set.

##### Returns
 Int
•

### Visualization

These property_types are for visualization.

•
Color

This is a pseudo-type for documentation purposes only. A function expecting an argument or option of type Color can digest an object of type RGB or HSV as well as a string with an RGB value in hex notation "#RRGGBB" or a symbolic color name.

•
Flexible

This is a pseudo-type for documentation purposes only. Many options of visualization functions modifying the appearance of some set of graphical elements like points, edges, facets, etc. accept a wide range of possible values, allowing for different grades of flexibility (and complexity):

SCALAR the same attribute value is applied to all elements ARRAY each element gets its individual attribute value HASH elements found in the hash get their individual attribute values, for the rest the appropriate default applies SUB a piece of code computing the attribute value for the given element

Unless specified explicitly in the detailed option description, the indices, keys, or subroutine arguments used for retrieval of the attribute values are just the zero-based ordinal numbers of the elements.

•
HSV

A color described as a Hue-Saturation-Value triple. Is convertible to and from an RGB representation.

•
RGB

A color described as a Red-Green-Blue triple. Can be constructed from a list of three integral values from the range 0..255, or a hex triplet with a leading # symbol, or a list of three floating-point values from the range 0..1, or a symbolic name from the X11 color names.

## Common Option Lists

•

### Visualization

These options are for visualization.

•
geometric_options

Options for visualizing objects with homogeneous coordinates like Polytope, PolyhedralComplex, SubdivisionOfPoints and PointConfiguration.

##### Options
 Matrix BoundingBox useful for unbounded polyhedra Matrix Transformation linear transformation, to be applied after dehomogenization Vector Offset shift, to be applied after dehomogenization and the linear transformation
•
geometric_options_linear

Options for visualizing objects with nonhomogeneous coordinates like Cone, PolyhedralFan and VectorConfiguration.

##### Options
 Matrix Transformation linear transformation, to be applied on rays/vectors
•
Visual::PointSet::decorations

Common attributes modifying the appearance of PointSets and all visual objects derived thereof. Please be aware that no one visualization program interfaced to polymake supports all of them. Unsupported options are normally ignored.

##### Options
 String Title the name of the drawing String Name the name of this visual object in the drawing Bool Hidden if set to true, the visual object is not rendered (useful for interactive visualization programs allowing for switching details on and off) String PointLabels if set to "hidden", no point labels are displayed String VertexLabels alias for PointLabels Flexible PointColor color of the spheres or rectangles representing the points Flexible VertexColor alias for PointColor Flexible PointThickness scaling factor for the size of the spheres or rectangles representing the points Flexible VertexThickness alias for PointThickness Flexible PointBorderColor color of the border line of rectangles representing the points Flexible VertexBorderColor alias for PointBorderColor Flexible PointBorderThickness scaling factor for the thickness of the border line of rectangles representing the points Flexible VertexBorderThickness alias for PointBorderThickness Flexible PointStyle if set to "hidden", neither point nor its label is rendered Flexible VertexStyle alias for PointStyle Vector ViewPoint ViewPoint for Sketch visualization Vector ViewDirection ViewDirection for Sketch visualization Vector ViewUp ViewUp for Sketch visualization Float Scale scale for Sketch visualization Flexible LabelAlignment Defines the alignment of the vertex labels: left, right or center
•
Visual::Polygon::decorations

Attributes modifying the appearance of filled polygons.

imports from: Visual::PointSet::decorations

##### Options
 Color FacetColor filling color of the polygon Float FacetTransparency transparency factor of the polygon between 0 (opaque) and 1 (completely translucent) String FacetStyle if set to "hidden", the inner area of the polygon is not rendered Color EdgeColor color of the boundary lines Float EdgeThickness scaling factor for the thickness of the boundary lines String EdgeStyle if set to "hidden", the boundary lines are not rendered
•
Visual::Polygons::decorations

Attributes modifying the appearance of a set of polygons (like a polygonal surface).

imports from: Visual::PointSet::decorations

##### Options
 Flexible FacetColor filling color of the polygons Flexible FacetTransparency transparency factor of the polygons between 0 (opaque) and 1 (completely translucent) Flexible FacetStyle if set to "hidden", the inner area of the polygons are not rendered at all String FacetLabels if set to "hidden", the facet labels are not displayed (in the most cases this is the default behavior) Color EdgeColor color of the boundary lines Float EdgeThickness scaling factor for the thickness of the boundary lines String EdgeStyle if set to "hidden", the boundary lines are not rendered
•
Visual::Wire::decorations

Attributes modifying the appearance of "wire frameworks". Unlike the rest, the flexible edge attributes are retrieved using the edge iterator as an index/key/argument.

imports from: Visual::PointSet::decorations

##### Options
 Flexible EdgeColor color of the lines representing the edges Flexible EdgeThickness scaling factor for the thickness of the lines representing the edges EdgeMap EdgeLabels textual labels to be placed along the edges Flexible EdgeStyle if set to "hidden", neither the edge nor its label is rendered