Isometric Embedding of Curvilinear Meshes Defined on Riemannian Metric Spaces

An algorithm for isometrically embedding curvilinear meshes defined on Riemannian metric spaces into Euclidean spaces of sufficiently high dimension is presented. The method is derived from the Landmark-Isomap algorithm and a previous method for embedding straight-sided meshes. The former is used to decrease the computational complexity of the embedding problem, notably the dense shortest-path problem used to estimate geodesic lengths across the mesh domain as well as the dense eigenvalue decomposition needed to compute the codimension coordinates. A method for defining curvilinear meshes from straight-sided ones in a dimension-independent manner is also discussed. Examples in two- and three-dimensions for both analytic embeddings and analytic metric fields are used to evaluate the method.