Software Open Access

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

Bartzos, Evangelos; Legerský, Jan

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 (3.5 MB)
Name Size
Computation_of_mixed_volume_and_complex_solutions.zip
md5:b0edcc899662c7f8664b039ff92f46e6
27.1 kB Download
Construction_of_Laman_and_Geiringer_graphs.zip
md5:7a101ac2d1a37f8b24dd4db5d5ba7c83
183.2 kB Download
Coupler_curve_visualization_and_sampling.zip
md5:db2d2a1435bdbe09f4c215d859bd1965
1.8 MB Download
Maple_parametric_search.zip
md5:6d9c95e3341a957883522116cd582440
34.9 kB Download
Results.zip
md5:7a8e6e1e95463ecf990d3ba6fec4d79b
1.5 MB Download
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