To compile it under linux: in compileAllMex.m set up the right paths for Eigen and libigl. Then to run the examples change the path formating to be linux-compatible: replace 'data\paper_deformation_2d_gecko_wConstraints.mat' by 'data/paper_deformation_2d_gecko_wConstraints.mat' for example (similar changes for all example scripts. Up to proper mesh visualization, the results are similar to the ones in the paper in both 2D and 3D.
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Please consider to cut/paste/edit the raw JSON data attached to this paper.