# Spatial graph embeddings and coupler curves¶

This program implements a method for obtaining edge lengths of a minimally rigid graph with many real spatial embeddings. The method is based on sampling over two parameter family that preserves so called coupler curve. See project website for the details.

Moreover, it includes Qt application for plotting coupler curves of the 7-vertex minimally rigid graph with the maximal number of embeddings, G48.

The main functionality is provided by the package *graphEmbeddings3D*, see Documentation.

The version 1.0 was used for the paper On the Maximal Number of Real Embeddings of Spatial Minimally Rigid Graphs accepted to ISSAC 2018:

The current version 2.0 supports arbitrary minimally rigid graphs containing a triangle instead of only predefined ones.

## Requirements and installation¶

- Python 2.7
- For solving the system of equations corresponding to graph embeddings, polynomial homotopy continuation by the package
`phcpy`

is used (homepages.math.uic.edu/~jan/phcpy_doc_html/). - In the sampling heuristic, clustering is done by
`DBSCAN`

from the package`sklearn`

(scikit-learn.org). - For GUI application for plotting coupler curves of G48,
`PyQt5`

(pypi.python.org/pypi/PyQt5) and`matplotlib`

(matplotlib.org/) are needed. - For installation, just clone or download from github.com/Legersky/SpatialGraphEmbeddings.

## Predefined graphs¶

- 6 vertices: octahedron/cyclohexane (the unique 6-vertex graph with the maximal number of embeddings)
- 7 vertices: G16a, G16b, G24, G32a, G32b, G48 (all 7-vertex graphs requiring the last Henneberg step being H2, the number corresponds to the number of embeddings)
- 8 vertices: G128, G160

## Tests¶

`python test_6vert.py`

runs the sampling method for octahedron

`python test_7vert.py`

verifies that there are edge lengths for G16a, G16b, G24, G32a, G32b and G48 such that all embeddings are real

`python test_8vert.py`

verifies that there are edge lengths such that G128 and G160 have 128, resp. 132, real embeddings

## Sampling¶

The scripts in the folder `sampling_scripts`

use the proposed method for various graphs and starting edge lengths.

## Coupler curves of G48¶

This Qt program is launched by `python CouplerCurveG48.py`

.

Functionality:

- loading and saving edge lengths
- plotting coupler curve of G48
- computing number of real embeddings of G48 by PHC
- sampling of parameters for specific subgraphs
- iterative method for increasing the number of real embeddings
- export to Axel

## Warning¶

The program strongly depends on PHC computation - this fails sometimes that might cause failure of the program.

## License¶

Copyright (C) 2018 Jan Legerský

This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version.

This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with this program. If not, see https://www.gnu.org/licenses/.