Restricted Constrained Delaunay Triangulations

We introduce the restricted constrained Delaunay triangulation (restricted CDT), a generalization of both the restricted Delaunay triangulation and the constrained Delaunay triangulation. The restricted CDT is a triangulation of a surface whose edges include a set of user-specified constraining segments. We define the restricted CDT to be the dual of a restricted Voronoi diagram defined on a surface that we have extended by topological surgery. We prove several properties of restricted CDTs, including sampling conditions under which the restricted CDT contains every constraining segment and is homeomorphic to the underlying surface

Simplified Emanation Graphs: A Sparse Plane Spanner with Steiner Points

Emanation graphs of grade $k$, introduced by Hamedmohseni, Rahmati, and Mondal, are plane spanners made by shooting $2^{k+1}$ rays from each given point, where the shorter rays stop the longer ones upon collision. The collision points are the Steiner points of the spanner. We introduce a method of simplification for emanation graphs of grade $k=2$, which makes it a competent spanner for many possible use cases such as network visualization and geometric routing. In particular, the simplification reduces the number of Steiner points by half and also significantly decreases the total number of edges, without increasing the spanning ratio. Exact methods of simplification along with mathematical proofs on properties of the simplified graph is provided. We compare simplified emanation graphs against Shewchuk's constrained Delaunay triangulations on both synthetic and real-life datasets. Our experimental results reveal that the simplified emanation graphs outperform constrained Delaunay triangulations in common quality measures (e.g., edge count, angular resolution, average degree, total edge length) while maintain a comparable spanning ratio and Steiner point count.Comment: A preliminary and shorter version of this paper was accepted in SOFSEM 202

Counting Triangulations and other Crossing-Free Structures Approximately

We consider the problem of counting straight-edge triangulations of a given set $P$ of $n$ points in the plane. Until very recently it was not known whether the exact number of triangulations of $P$ can be computed asymptotically faster than by enumerating all triangulations. We now know that the number of triangulations of $P$ can be computed in $O^{*}(2^{n})$ time, which is less than the lower bound of $\Omega(2.43^{n})$ on the number of triangulations of any point set. In this paper we address the question of whether one can approximately count triangulations in sub-exponential time. We present an algorithm with sub-exponential running time and sub-exponential approximation ratio, that is, denoting by $\Lambda$ the output of our algorithm, and by $c^{n}$ the exact number of triangulations of $P$, for some positive constant $c$, we prove that $c^{n}\leq\Lambda\leq c^{n}\cdot 2^{o(n)}$. This is the first algorithm that in sub-exponential time computes a $(1+o(1))$-approximation of the base of the number of triangulations, more precisely, $c\leq\Lambda^{\frac{1}{n}}\leq(1 + o(1))c$. Our algorithm can be adapted to approximately count other crossing-free structures on $P$, keeping the quality of approximation and running time intact. In this paper we show how to do this for matchings and spanning trees.Comment: 19 pages, 2 figures. A preliminary version appeared at CCCG 201

Detecting and decomposing self-overlapping curves

AbstractA curve is self-overlapping if it can be divided by nontrivial line segments into simple curves. We show how to test whether a given curve is self-overlapping, and also how to construct sets of line segments that demonstrate that the curve is self-overlapping. We also describe several interesting topological properties of self-overlapping curves

Finding the constrained Delaunay triangulation and constrained Voronoi diagram of a simple polygon in linear time

In this paper, we present a Θ(n) time worst-case deterministic algorithm for finding the constrained Delaunay triangulation and constrained Voronoi diagram of a simple n-sided polygon in the plane. Up to now, only an O(n log n) worst-case deterministic and an O(n) expected time bound have been shown, leaving an O(n) deterministic solution open to conjecture.published_or_final_versio

Computing Three-dimensional Constrained Delaunay Refinement Using the GPU

We propose the first GPU algorithm for the 3D triangulation refinement problem. For an input of a piecewise linear complex $\mathcal{G}$ and a constant $B$, it produces, by adding Steiner points, a constrained Delaunay triangulation conforming to $\mathcal{G}$ and containing tetrahedra mostly of radius-edge ratios smaller than $B$. Our implementation of the algorithm shows that it can be an order of magnitude faster than the best CPU algorithm while using a similar amount of Steiner points to produce triangulations of comparable quality