15 Result(s)
-
Article
Open AccessLocal Criteria for Triangulating General Manifolds
We present criteria for establishing a triangulation of a manifold. Given a manifold M, a simplicial complex \({\mathscr {A}}\) ...
-
Article
Open AccessLocal Conditions for Triangulating Submanifolds of Euclidean Space
We consider the following setting: suppose that we are given a manifold M in \({\mathbb {R}}^d\) ...
-
Article
Open AccessRandomized Incremental Construction of Delaunay Triangulations of Nice Point Sets
Randomized incremental construction (RIC) is one of the most important paradigms for building geometric data structures. Clarkson and Shor developed a general theory that led to numerous algorithms which are both...
-
Article
Open AccessTriangulating Submanifolds: An Elementary and Quantified Version of Whitney’s Method
We quantise Whitney’s construction to prove the existence of a triangulation for any \(C^2\) ...
-
Article
An Obstruction to Delaunay Triangulations in Riemannian Manifolds
Delaunay has shown that the Delaunay complex of a finite set of points \(P\) ...
-
Article
Manifold Reconstruction Using Tangential Delaunay Complexes
We give a provably correct algorithm to reconstruct a k-dimensional smooth manifold embedded in d-dimensional Euclidean space. The input to our algorithm is a point sample coming from an unknown manifold. Our app...
-
Article
Geometric Tomography with Topological Guarantees
We consider the problem of reconstructing a compact 3-manifold (with boundary) embedded in \(\mathbb R ^3\) ...
-
Article
Bregman Voronoi Diagrams
The Voronoi diagram of a finite set of objects is a fundamental geometric structure that subdivides the embedding space into regions, each region consisting of the points that are closer to a given object than...
-
Article
Manifold Reconstruction in Arbitrary Dimensions Using Witness Complexes
It is a well-established fact that the witness complex is closely related to the restricted Delaunay triangulation in low dimensions. Specifically, it has been proved that the witness complex coincides with th...
-
Article
Isotopic Implicit Surface Meshing
This paper addresses the problem of piecewise linear approximation of implicit surfaces. We first give a criterion ensuring that the zero-set of a smooth function and the one of a piecewise linear approximati...
-
Article
A Linear Bound on the Complexity of the Delaunay Triangulation of Points on Polyhedral Surfaces
Delaunay triangulations and Voronoi diagrams have found numerous applications in surface modeling, surface mesh generation, deformable surface modeling and surface reconstruction. Many algorithms i...
-
Article
Complexity of the Delaunay Triangulation of Points on Polyhedral Surfaces
It is well known that the complexity of the Delaunay triangulation of $n$ points in $\RR ^d$, i.e., the number of its simplices, can be $\Omega (n^{\lceil {d}/{2}\rceil })$. In particular, in $\RR ^3$, the nu...
-
Chapter and Conference Paper
Convex tours of bounded curvature
We consider the motion planning problem for a point constrained to move along a smooth closed convex path of bounded curvature. The workspace of the moving point is bounded by a convex polygon with m vertices, co...
-
Article
Applications of random sampling to on-line algorithms in computational geometry
This paper presents a general framework for the design and randomized analysis of geometric algorithms. These algorithms are on-line and the framework provides general bounds for their expected space and time ...
-
Chapter and Conference Paper
Polygon placement under translation and rotation
