Ambient Isotopic Meshing of Implicit Algebraic Surface with Singularities

A complete method is proposed to compute a certified, or ambient isotopic, meshing for an implicit algebraic surface with singularities. By certified, we mean a meshing with correct topology and any given geometric precision. We propose a symbolic-numeric method to compute a certified meshing for the surface inside a box containing singularities and use a modified Plantinga-Vegter marching cube method to compute a certified meshing for the surface inside a box without singularities. Nontrivial examples are given to show the effectiveness of the algorithm. To our knowledge, this is the first method to compute a certified meshing for surfaces with singularities.Comment: 34 pages, 17 Postscript figure

The Complexity of Subdivision for Diameter-Distance Tests

We present a general framework for analyzing the complexity of subdivision-based algorithms whose tests are based on the sizes of regions and their distance to certain sets (often varieties) intrinsic to the problem under study. We call such tests diameter-distance tests. We illustrate that diameter-distance tests are common in the literature by proving that many interval arithmetic-based tests are, in fact, diameter-distance tests. For this class of algorithms, we provide both non-adaptive bounds for the complexity, based on separation bounds, as well as adaptive bounds, by applying the framework of continuous amortization. Using this structure, we provide the first complexity analysis for the algorithm by Plantinga and Vegeter for approximating real implicit curves and surfaces. We present both adaptive and non-adaptive a priori worst-case bounds on the complexity of this algorithm both in terms of the number of subregions constructed and in terms of the bit complexity for the construction. Finally, we construct families of hypersurfaces to prove that our bounds are tight

The topological correctness of PL approximations of isomanifolds

Isomanifolds are the generalization of isosurfaces to arbitrary dimension and codimension, i.e. manifolds defined as the zero set of some multivariate vector-valued smooth function f : Rd → Rd−n. A natural (and efficient) way to approximate an isomanifold is to consider its Piecewise-Linear (PL) approximation based on a triangulation T of the ambient space Rd. In this paper, we give conditions under which the PL-approximation of an isomanifold is topologically equivalent to the isomanifold. The conditions are easy to satisfy in the sense that they can always be met by taking a sufficiently fine triangulation T . This contrasts with previous results on the triangulation of manifolds where, in arbitrary dimensions, delicate perturbations are needed to guarantee topological correctness, which leads to strong limitations in practice. We further give a bound on the Fréchet distance between the original isomanifold and its PL-approximation. Finally we show analogous results for the PL-approximation of an isomanifold with boundary

Isotopic triangulation of a real algebraic surface

International audienceWe present a new algorithm for computing the topology of a real algebraic surface $S$ in a ball $B$, even in singular cases. We use algorithms for 2D and 3D algebraic curves and show how one can compute a topological complex equivalent to $S$, and even a simplicial complex isotopic to $S$ by exploiting properties of the contour curve of $S$. The correctness proof of the algorithm is based on results from stratification theory. We construct an explicit Whitney stratification of $S$, by resultant computation. Using Thom's isotopy lemma, we show how to deduce the topology of $S$ from a finite number of characteristic points on the surface. An analysis of the complexity of the algorithm and effectiveness issues conclude the paper

Automatic Linear and Curvilinear Mesh Generation Driven by Validity Fidelity and Topological Guarantees

Image-based geometric modeling and mesh generation play a critical role in computational biology and medicine. In this dissertation, a comprehensive computational framework for both guaranteed quality linear and high-order automatic mesh generation is presented. Starting from segmented images, a quality 2D/3D linear mesh is constructed. The boundary of the constructed mesh is proved to be homeomorphic to the object surface. In addition, a guaranteed dihedral angle bound of up to 19:47o for the output tetrahedra is provided. Moreover, user-specified guaranteed bounds on the distance between the boundaries of the mesh and the boundaries of the materials are allowed. The mesh contains a small number of mesh elements that comply with these guarantees, and the runtime is compatible in performance with other software. Then the curvilinear mesh generator allows for a transformation of straight-sided meshes to curvilinear meshes with C1 or C2 smooth boundaries while keeping all elements valid and with good quality as measured by their Jacobians. The mathematical proof shows that the meshes generated by our algorithm are guaranteed to be homeomorphic to the input images, and all the elements inside the meshes are guaranteed to be with good quality. Experimental results show that the mesh boundaries represent the objects\u27 shapes faithfully, and the accuracy of the representation is improved compared to the corresponding linear mesh