Graph embeddings in the plane, space and sphere - source code and results

2018-11-24T00:00:00Z (GMT) by Bartzos, Evangelos Legerský, Jan
<p>This is supplementary material for the paper <em>On the Maximal Number of Real Embeddings of Minimally Rigid Graphs in \(\mathbb{R}^2\), \(\mathbb{R}^3\) and \(S^2\)</em>. It includes source code and results. See also the <a href="http://jan.legersky.cz/project/spatialgraphembeddings/">project website</a>.</p> <p><strong>Construction of Laman and Geiringer graphs</strong></p> <p>MATLAB and SageMath code for construction of Laman and Geiringer graphs using Henneberg steps.</p> <p><strong>Computation of mixed volume and complex solutions</strong></p> <p>Functions that compute the mixed volume and the (complex) embeddings in the plane, space or sphere of a given graph using different algebraic formulations.</p> <p><strong>Coupler curve visualization and sampling method</strong></p> <p>Implementation of our method for obtaining edge lengths of a Geiringer graph with many real spatial embeddings.<br> Visualization of coupler curves of the 7-vertex Geiringer graph with the maximal number of real embeddings (G48).</p> <p><strong>Maple parametric search</strong></p> <p>Maple worksheet illustrating a method to improve the number of real embeddings of G48 using RootFinding[Parametric] subpackage.</p> <p><strong>Results</strong></p> <p>Mixed volumes and numbers of real and complex embeddings in the plane, space and sphere.<br> Edge lengths giving high numbers of real embeddings, in the plane, space and sphere. <br> Maple script verifying the results.</p>