Properties of n-dimensional triangulations

A Local Convergence Proof for the Minvar Algorithm for Computing Continuous Piecewise Linear Approximations

“…These heuristics flip edges in the domain triangulation to improve a local goodness criterion, similar to a method for computing the planar Delaunay triangulation. Generalizing these heuristics to higher dimensions is difficult because local topological changes of the triangulation in dimensions greater than two are more complex than edge flipping [28,14]. Nonetheless, we find that adaptation of the combinatorial parameters of the PL function via topological flipping provides significant benefit in practice.…”

“…Not all combinations of continuous and combinatorial parameters yield a proper triangulation. Triangulations in two and three dimensions have been studied extensively in the computational geometry literature [13,31], but results for general dimension are more scarce, notwithstanding significant recent progress [5,14,28]. The price of using a family of finitely parameterized homeomorphisms, the PL approximations, is the cost of managing the combinatorial complexities of PL functions.…”

Geometric bistellar flips: the setting, the context and a construction

“…Lawson himself, in 1986 [42], is close to defining flips in arbitrary dimension, even in the case of special position. Around 1990, 17 B. Joe realizes that in dimension three one cannot, in general, monotonically flip from any triangulation to the Delaunay triangulation [37] but, still, the following incremental algorithm works [38]: insert the points one by one, each by an insertion flip in the Delaunay triangulation of the already inserted points.…”

Efficient surface reconstruction from contours based on two‐dimensional Delaunay triangulation

“…Two approaches can be followed to obtain an updated Delaunay triangulation T new after the insertion of a new point P into an existing Delaunay triangulation T old [22,[25][26][27]. The approach adopted in the present work uses local splitting together with repeated edge swapping until all the edges, except constrained edges, satisfy the Delaunay property [25,26].…”

“…Oliva et al [3], constructed the Voronoi diagram and then subdivided each cell into triangles. However, as the well-developed Delaunay triangulation [22][23][24][25][26] provides the optimality of nearest connection, and this in turn favours vertical connection in the reconstruction, the problems of tiling and correspondence are naturally solved.…”