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"... ἀγεωµὲρητος µηδεὶς εἰσίτω Let no one ignorant of geometry enter here Engraved at the door of Plato’s Academy in ancient Athens ..."

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ἀγεωµὲρητος µηδεὶς εἰσίτω Let no one ignorant of geometry enter here Engraved at the door of Plato’s Academy in ancient Athens

### Efficient Construction and Simplification of Delaunay Meshes Yong-Jin Liu∗

"... Figure 1: We present an efficient algorithm to convert an arbitrary manifold triangle meshM to a Delaunay mesh (DM), which has the same geometry of M. Our algorithm can also produce progressive Delaunay meshes, allowing a smooth choice of detail levels. Since DMs are represented using conventional m ..."

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Figure 1: We present an efficient algorithm to convert an arbitrary manifold triangle meshM to a Delaunay mesh (DM), which has the same geometry of M. Our algorithm can also produce progressive Delaunay meshes, allowing a smooth choice of detail levels. Since DMs are represented using conventional mesh data structures, the existing digital geometry processing algorithms can benefit the numerical stability of DM without changing any codes. For example, DMs significantly improve the accuracy of the heat method for computing geodesic distances. Delaunay meshes (DM) are a special type of triangle mesh where the local Delaunay condition holds everywhere. We present an efficient algorithm to convert an arbitrary manifold triangle mesh M into a Delaunay mesh. We show that the constructed DM has O(Kn) vertices, where n is the number of vertices in M and K is a model-dependent constant. We also develop a novel algorithm to simplify Delaunay meshes, allowing a smooth choice of detail levels. Our methods are conceptually simple, theoretically sound and easy to implement. The DM construction algorithm also scales well due to its O(nK logK) time complexity.

### Anisotropic Simplicial Meshing Using Local Convex Functions

"... Figure 1: Anisotropic meshes generated by our method. Left: 2D meshing. The anisotropic metric is defined as the Hessian of an analytic function evincing a large range of anisotropy ratios (λ ∈ [1.9, 394.4]). Zoom in on the image to see meshing details. Middle: 3D surface meshing of the Happy Buddha ..."

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Figure 1: Anisotropic meshes generated by our method. Left: 2D meshing. The anisotropic metric is defined as the Hessian of an analytic function evincing a large range of anisotropy ratios (λ ∈ [1.9, 394.4]). Zoom in on the image to see meshing details. Middle: 3D surface meshing of the Happy Buddha from curvature tensors estimated from a high-resolution reference mesh (115474 vertices). Our relaxation produces a high quality result (63284 vertices) starting with an initial low-resolution mesh (5000 vertices). Right: volumetric meshing in a 3D cube. Anisotropy changes substantially (λ ∈ [1, 40]) and rapidly over the domain. The lower image shows a cross-section. We present a novel method to generate high-quality simplicial meshes with specified anisotropy. Given a surface or volumetric domain equipped with a Riemannian metric that encodes the desired anisotropy, we transform the problem to one of functional approx-imation. We construct a convex function over each mesh simplex whose Hessian locally matches the Riemannian metric, and itera-tively adapt vertex positions and mesh connectivity to minimize the difference between the target convex functions and their piecewise-linear interpolation over the mesh. Our method generalizes optimal Delaunay triangulation and leads to a simple and efficient algorithm. We demonstrate its quality and speed compared to state-of-the-art methods on a variety of domains and metrics.