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In 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 <inline-formula> <math display="inline"> <semantics> <mi>ϕ</mi> </semantics> </math> </inline-formula> from the original mesh <inline-formula> <math display="inline"> <semantics> <mrow> <mi>M</mi> <mo>&#8712;</mo> <msup> <mi mathvariant="double-struck">R</mi> <mn>3</mn> </msup> </mrow> </semantics> </math> </inline-formula> to the planar domain <inline-formula> <math display="inline"> <semantics> <mrow> <mi>ϕ</mi> <mrow> <mo>(</mo> <mi>M</mi> <mo>)</mo> </mrow> <mo>&#8712;</mo> <msup> <mi mathvariant="double-struck">R</mi> <mn>2</mn> </msup> </mrow> </semantics> </math> </inline-formula>. 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 <i>M</i>. 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 <i>M</i> and Euclidean distances on <inline-formula> <math display="inline"> <semantics> <msup> <mi mathvariant="double-struck">R</mi> <mn>2</mn> </msup> </semantics> </math> </inline-formula>. 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.