2021-04-162019-01-0122277390WOS;000482856500100SCOPUS;2-s2.0-85071185884http://hdl.handle.net/10784/29550In the context of CAD, CAM, CAE, and reverse engineering, the problem of mesh parameterization is a central process. Mesh parameterization implies the computation of a bijective map ? from the original mesh M ? R3 to the planar domain ?(M) ? R2. The mapping may preserve angles, areas, or distances. Distance-preserving parameterizations (i.e., isometries) are obviously attractive. However, geodesic-based isometries present limitations when the mesh has concave or disconnected boundary (i.e., holes). Recent advances in computing geodesic maps using the heat equation in 2-manifolds motivate us to revisit mesh parameterization with geodesic maps. We devise a Poisson surface underlying, extending, and filling the holes of the mesh M. We compute a near-isometric mapping for quasi-developable meshes by using geodesic maps based on heat propagation. Our method: (1) Precomputes a set of temperature maps (heat kernels) on the mesh; (2) estimates the geodesic distances along the piecewise linear surface by using the temperature maps; and (3) uses multidimensional scaling (MDS) to acquire the 2D coordinates that minimize the difference between geodesic distances on M and Euclidean distances on R2. This novel heat-geodesic parameterization is successfully tested with several concave and/or punctured surfaces, obtaining bijective low-distortion parameterizations. Failures are registered in nonsegmented, highly nondevelopable meshes (such as seam meshes). These cases are the goal of future endeavors. © 2019 by the authors.https://v2.sherpa.ac.uk/id/publication/issn/2227-7390info:eu-repo/semantics/articleGeodesic mapsHeat transfer analysisMesh parameterizationPoisson fills2021-04-16Mejia-Parra D.Sánchez J.R.Posada J.Ruiz-Salguero O.Cadavid C.10.3390/math7080753