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.

Requirements and installation

Supported 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


If you want to compute the number of embeddings for another minimally rigid graph, please, provide a method constructEquations_YOUR_GRAPH with sphere equations in graphEmbeddings3D.graphEmbedding, and modify the constructor accordingly. For the sampling method, subgraphs suitable for sampling must be added to constructor of graphEmbeddings3D.algRealEmbeddings. We appreciate if you share your changes in the GitHub repository.


python runs the sampling method for octahedron

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

python verifies that there are edge lengths G128 and G160 have 128 real embeddings


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


  • 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


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


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