Graph embeddings in the plane, space and sphere - source code and results
Creators
- 1. Department of Informatics and Telecommunications, National Kapodistrian University of Athens, and ATHENA Research Center
- 2. Research Institute for Symbolic Computation, Johannes Kepler University Linz, Austria
Description
This is supplementary material for the paper On the Maximal Number of Real Embeddings of Minimally Rigid Graphs in \(\mathbb{R}^2\), \(\mathbb{R}^3\) and \(S^2\). It includes source code and results. See also the project website. The material extends the results of our conference paper (ISSAC 2018).
Construction of Laman and Geiringer graphs
MATLAB and SageMath code for construction of Laman and Geiringer graphs using Henneberg steps.
Computation of mixed volume and complex solutions
Functions that compute the mixed volume and the (complex) embeddings in the plane, space or sphere of a given graph using different algebraic formulations.
Coupler curve visualization and sampling method
Implementation of our method for obtaining edge lengths of a Geiringer graph with many real spatial embeddings.
Visualization of coupler curves of the 7-vertex Geiringer graph with the maximal number of real embeddings (G48).
Maple parametric search
Maple worksheet illustrating a method to improve the number of real embeddings of G48 using RootFinding[Parametric] subpackage.
Results
Mixed volumes and numbers of real and complex embeddings in the plane, space and sphere.
Edge lengths giving high numbers of real embeddings, in the plane, space and sphere.
Maple script verifying the results.
Files
Computation_of_mixed_volume_and_complex_solutions.zip
Files
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