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Generalized Triangulations
Published in Kai Hormann, N. Sukumar, Generalized Barycentric Coordinates in Computer Graphics and Computational Mechanics, 2017
A triangle mesh is a partition of a bounded domain with simplices (triangles in 2D, tetrahedra in 3D) so that any two of these simplices are either disjoint or sharing a face. The resulting triangulation (also called tetrahedralizations in 3D) provides a discretization of space through both its primal (simplicial) elements andits dual (cell) elements. Both types of element are crucial to a variety of numerical techniques. A growing trend in numerical simulation is the simultaneous use of primal and dual meshes: Petrov–Galerkin finite element/finite volume methods [24,271,298] and methods based on exterior calculus [65,119,176] use the ability to store quantities on both primal and dual elements to enforce (co)homological relationships, which for instance appear in Hodge theory. The choice of the dual, defined by the location of the dual vertices, is however not specified a priori. The barycentric dual, for which barycenters are used instead of circumcenters, is used for certain finite volume computations, but it fails to satisfy both the orthogonality and the convexity conditions on general triangulations. A very common dual to a triangulation in Rd $ \mathbb R ^d $ is the cell complex that uses the circumcenters of each d-simplex as dual vertices. It corresponds to the Voronoi diagram in the case of Delaunay triangulations [137,313]. The circumcentric Delaunay–Voronoi duality has been extensively used in diverse fields. Building on a number of results in algebraic and computational geometry, a more general primal-dual pair of complexes can be defined [274], as we briefly describe in the first sections of this chapter. A barycentric representation of primal-dual triangulations is then presented as a parameterization of this space of generalized triangulations.
Machined sharp edge restoration for triangle mesh workpiece models derived from grid-based machining simulation
Published in Computer-Aided Design and Applications, 2018
Ziqi Wang, Jack Szu-Shen Chen, Jimin Joy, Hsi-Yung Feng
Triangle mesh is a versatile geometric representation format for machining simulation as it provides simple visualization and surface analysis functionalities. Geometric machining simulation is an essential part of the emerging virtual machining technology [2]. Compared to other representation formats, triangle mesh can represent the workpiece shape at reasonable accuracy with relatively low model size. This is achievable since triangle mesh uses simple triangular elements to approximate the workpiece surface geometry and a large number of smaller triangles can be used at areas of high curvature and a lesser number of large triangles at flatter areas. It makes triangle mesh a simple and efficient format for visualization and surface analysis in machining simulation.
A triangle mesh-based corner detection algorithm for catadioptric images
Published in The Imaging Science Journal, 2018
A triangle mesh is a discrete structure that can be used to approximate a surface in Euclidean space [22]. In geometry processing algorithms, triangle meshes are considered a collection of triangles, which define segments of a linear surface via barycentric parameterization. Triangle mesh is generated by regularly splitting faces, starting from an initial coarse triangulation (Figure 2). At each iteration, each triangle is splitted into four sub-triangles. Hence, the approximation error of a triangle mesh is inversely proportional to its number of triangles.