Available versions of this document: latest release, release 3.5, nightly master
Reference documentation for older polymake versions: release 3.4, release 3.3, release 3.2
BigObject RationalCurve
from application tropical
An nmarked rational curve, identified by its SETS
, i.e. its partitions of {1,…,n} and its COEFFICIENTS
, i.e. the lengths of the corresponding edges.
Properties
Input property
These properties are for input only. They allow redundant information.

INPUT_COEFFS
Same as
COEFFS
, except that entries may be ⇐0. This should have the same length asINPUT_SETS
. Type:
 from extension:

INPUT_SETS
Same as
SETS
, except that sets may appear several times. Type:
 from extension:

INPUT_STRING
This property can also be used to define a rational curve: A linear combination of partitions is given as a string, using the following syntax: A partition is given as a subset of {1,..,n} and written as a commaseparated list of leaf indices in round brackets, e.g. “(1,2,5)” A linear combination can be created using rational numbers, “+”,“+” and “” in the obvious way, e.g. “2*(1,2,5) + 1*(3,4,7)  2(1,2) (The “*” is optional) Of course, each set should contain at least two elements. If you don't specify N_LEAVES, it is set to be the largest leaf index occuring in the sets. Partitions needn't be irredundant and coefficients can be any rational number. If the resulting element is not in the moduli space, an error is thrown.
 Type:
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Combinatorics
These properties capture combinatorial information of the object. Combinatorial properties only depend on combinatorial data of the object like, e.g., the face lattice.

COEFFS
A list of positive rational coefficients. The list should have the same length as
SETS
and contain only entries > 0. The ith entry then gives the length of the bounded edge defined by the ith partition. If you're not sure if all your coefficients are > 0, useINPUT_SETS
andINPUT_COEFFS
instead. Note that the zero curve (i.e. no bounded edges, only leaves) is represented by one empty set with corresponding lenghth 0. Type:
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GRAPH
Contains the abstract graph (nonmetric) corresponding to the curve. All unbounded leaves are modelled as bounded edges. The vertices at the ends of the “leaves” are always the first
N_LEAVES
vertices. Type:
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GRAPH_EDGE_LENGTHS
Contains the lengths of the edges of
GRAPH
that represent bounded edges of the curve. The coefficients appear in the order that the corr. edges appear inEDGES
. Type:
 from extension:

NODES_BY_LEAVES
This incidence matrix gives a list of the vertices of the curve Each row corresponds to a vertex and contains as a set the leaves that are attached to that vertex (again, counting from 1!)
 Type:
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NODES_BY_SETS
This incidence matrix gives a list of the vertices of the curve Each row corresponds to a vertex and contains as a set the row indices of the
SETS
that correspond to edges attached to that vertex Type:
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NODE_DEGREES
This gives a list of the vertices of the curve in terms of their valences They appear in the same order as in
NODES_BY_LEAVES
orNODES_BY_SETS
 Type:
 from extension:

N_LEAVES
The number of leaves of the rational curve.
 Type:
 from extension:

SETS
A list of partitions of [n] that define the tree of the curve: For each bounded edge we have the corresponding partition of the n leaves. These should be irredundant. If you want to input a possibly redundant list, use
INPUT_SETS
andINPUT_COEFFS
instead. The number of marked leaves should always be given byN_LEAVES
. The sets are subsets of {1,…,n} (NOT {0,..,n1}!) Note that the zero curve (i.e. no bounded edges, only leaves) is represented by one empty set with corresponding lenghth 0. Type:
 from extension:
Methods
Conversion
These deal with converting the representation of a rational curve between metric vector and matroid fan coordinates.

metric_vector
Returns the (n over 2) metric vector of the rational nmarked curve
 from extension:
Visualization
These methods are for visualization.

VISUAL()
Visualizes a RationalCurve object. This visualization uses the VISUAL method of its GRAPH, so it accepts all the options of Visual::Graph::decorations. In addition it has another option
 Options:
String
LengthLabels
: If “hidden”, the edges are not labelled with their lengths. Any other text is ignored. Not set to “hidden” by default. option list
Visual::RationalCurve::decorations
 from extension: