Simple Polygon Research Articles

Computing the geodesic centers of a polygonal domain

We present an algorithm that computes the geodesic center of a given polygonal domain. The running time of our algorithm is O(n12+ϵ) for any ϵ>0, where n is the number of corners of the input polygonal domain. Prior to our work, only the very special case where a simple polygon is given as input has been intensively studied in the 1980s, and an O(nlog⁡n)-time algorithm is known by Pollack et al. Our algorithm is the first one that can handle general polygonal domains having one or more polygonal holes.

A linear time algorithm for minimum conic link path in a simple polygon

We consider the problem of finding, in a simple polygon, from a starting point to a destination point, a piecewise path consisting of conic sections. By considering only one type of conic section, i.e., circular, elliptic, hyperbolic, or parabolic curves, we present an O(n) time algorithm for computing the path with the minimum number of conic sections. The studied problem is the generalization of the straight line link path version. The results can be conducted in versatile applications: the hidden surface removal problem in CAD/CAM, the contour tracing, the red-blue intersection problem, the robot motion planning, and related computational geometry applications. The linear time property is most vital for those applications need to take instant reaction.

Near optimal line segment queries in simple polygons

This paper considers the problem of computing the weak visibility polygon (WVP) of any query line segment pq (or WVP(pq)) inside a given simple polygon P. We present an algorithm that preprocesses P and creates a data structure from which WVP(pq) is efficiently reported in an output sensitive manner.Our algorithm needs O(n2log⁡n) time and O(n2) space in the preprocessing phase to report WVP(pq) of any query line segment pq in time O(|WVP(pq)|+log2⁡n+κlog2⁡(nκ)), where κ is an input and output sensitive parameter of at most |WVP(pq)|. We improve the preprocessing time and space of current results for this problem [11,6] at the expense of more query time.

Efficient Multi-Robot Motion Planning for Unlabeled Discs in Simple Polygons

We consider the following motion-planning problem: we are given $ \mbi{m} $ unit discs in a simple polygon with $ \mbi{n} $ vertices, each at their own start position, and we want to move the discs to a given set of $ \mbi{m} $ target positions. Contrary to the standard (labeled) version of the problem, each disc is allowed to be moved to any target position, as long as in the end every target position is occupied. We show that this unlabeled version of the problem can be solved in $ \mbi{O(m^{2}+mn)} $ time, assuming that the start and target positions are at least some minimal distance from each other. This is in sharp contrast to the standard (labeled) and more general multi-robot motion planning problem for discs moving in a simple polygon, which is known to be strongly NP-hard.

Tracking Pavement Cells through Space and Time: Microtubules Define Positions of Lobe Formation.

Pavement cells in the leaf epidermis start out as simple polygons and develop into puzzle-shaped units with interlocking lobes. This complex geometry offers biomechanical advantages; interlocking increases resistance to tension and maintains the structural integrity of the leaf surface ([Malinowski

Exploring Curved Schematization of Territorial Outlines.

Hand-drawn schematized maps traditionally make extensive use of curves. However, there are few automated approaches for curved schematization; most previous work focuses on straight lines. We present a new algorithm for area-preserving curved schematization of territorial outlines. Our algorithm converts a simple polygon into a schematic crossing-free representation using circular arcs. We use two basic operations to iteratively replace consecutive arcs until the desired complexity is reached. Our results are not restricted to arcs ending at input vertices. The method can be steered towards different degrees of "curviness": we can encourage or discourage the use of arcs with a large central angle via a single parameter. Our method creates visually pleasing results even for very low output complexities. To evaluate the effectiveness of our design choices, we present a geometric evaluation of the resulting schematizations. Besides the geometric qualities of our algorithm, we also investigate the potential of curved schematization as a concept. We conducted an online user study investigating the effectiveness of curved schematizations compared to straight-line schematizations. While the visual complexity of curved shapes was judged higher than that of straight-line shapes, users generally preferred curved schematizations. We observed that curves significantly improved the ability of users to match schematized shapes of moderate complexity to their unschematized equivalents.

Diffuse Reflection Radius in a Simple Polygon

It is shown that every simple polygon in general position with n walls can be illuminated from a single point light source s after at most $$\lfloor (n-2)/4\rfloor $$ diffuse reflections, and this bound is the best possible. A point s with this property can be computed in $$O(n \log n)$$ time. It is also shown that the minimum number of diffuse reflections needed to illuminate a given simple polygon from a single point can be approximated up to an additive constant in polynomial time.

Flip Distance Between Triangulations of a Simple Polygon is NP-Complete

Let T be a triangulation of a simple polygon. A flip in T is the operation of removing one diagonal of T and adding a different one such that the resulting graph is again a triangulation. The flip distance between two triangulations is the smallest number of flips required to transform one triangulation into the other. For the special case of convex polygons, the problem of determining the shortest flip distance between two triangulations is equivalent to determining the rotation distance between two binary trees, a central problem which is still open after over 25 years of intensive study. We show that computing the flip distance between two triangulations of a simple polygon is NP-complete. This complements a recent result that shows APX-hardness of determining the flip distance between two triangulations of a planar point set.

Diffuse reflection diameter in simple polygons

We prove a conjecture of Aanjaneya, Bishnu, and Pal that the minimum number of diffuse reflections sufficient to illuminate the interior of any simple polygon with n walls from any interior point light source is ⌊n/2⌋−1. Light reflecting diffusely leaves a surface in all directions, rather than at an identical angle as with specular reflections.

Weak visibility counting in simple polygons

For a simple polygon P of size n, we define weak visibility counting problem (WVCP) as finding the number of visible segments of P from a query line segment pq. We present different algorithms to compute WVCP in sub-linear time. In our first algorithm, we spend O(n7) time to preprocess the polygon and build a data structure of size O(n6), so that we can optimally answer WVCP in O(logn) time. Then, we reduce the preprocessing costs to O(n4+ϵ) time and space at the expense of more query time of O(log5n). We also obtain a trade-off between preprocessing and query time costs. Finally, we propose an approximation method to reduce the preprocessing costs to O(n2) time and space and O(n1/2+ϵ) query time.

GPU-based parallel algorithm for computing point visibility inside simple polygons

Given a simple polygon P in the plane, we present a parallel algorithm for computing the visibility polygon of an observer point q inside P. We use chain visibility concept and a bottom-up merge method for constructing the visibility polygon of point q. The algorithm is simple and mainly designed for GPU architectures, where it runs in O(logn) time using O(n) processors. This is the first work on designing a GPU-based parallel algorithm for the visibility problem. To the best of our knowledge, the presented algorithm is also the first suboptimal parallel algorithm for the visibility problem that can be implemented on existing parallel architectures. We evaluated a sample implementation of the algorithm on several large polygons. The experiments indicate that our algorithm can compute the visibility of a polygon having over 4M points in tenth of a second on an NVIDIA GTX 780 Ti GPU.

Weak visibility queries of line segments in simple polygons

Given a simple polygon P in the plane, we present new data structures for computing the weak visibility polygon from any query line segment in P. We build a data structure in O(n) time and space that can compute the visibility polygon for any query line segment s in O(klog⁡n) time, where k is the size of the visibility polygon of s and n is the number of vertices of P. Alternatively, we build a data structure in O(n3) time and space that can compute the visibility polygon for any query line segment in O(k+log⁡n) time. In order to develop these data structures, we obtain many other results that may be interesting in their own right.

Computing the [formula omitted] geodesic diameter and center of a simple polygon in linear time

In this paper, we show that the L1 geodesic diameter and center of a simple polygon can be computed in linear time. For the purpose, we focus on revealing basic geometric properties of the L1 geodesic balls, that is, the metric balls with respect to the L1 geodesic distance. More specifically, in this paper we show that any family of L1 geodesic balls in any simple polygon has Helly number two, and the L1 geodesic center consists of midpoints of shortest paths between diametral pairs. These properties are crucial for our linear-time algorithms, and do not hold for the Euclidean case.

A novel integrated chassis controller for full drive-by-wire vehicles

In this paper, a systematic design with multiple hierarchical layers is adopted in the integrated chassis controller for full drive-by-wire vehicles. A reference model and the optimal preview acceleration driver model are utilised in the driver control layer to describe and realise the driver's anticipation of the vehicle's handling characteristics, respectively. Both the sliding mode control and terminal sliding mode control techniques are employed in the vehicle motion control (MC) layer to determine the MC efforts such that better tracking performance can be attained. In the tyre force allocation layer, a polygonal simplification method is proposed to deal with the constraints of the tyre adhesive limits efficiently and effectively, whereby the load transfer due to both roll and pitch is also taken into account which directly affects the constraints. By calculating the motor torque and steering angle of each wheel in the executive layer, the total workload of four wheels is minimised during normal driving, whereas the MC efforts are maximised in extreme handling conditions. The proposed controller is validated through simulation to improve vehicle stability and handling performance in both open- and closed-loop manoeuvres.

Constrained Point Set Embedding of a Balanced Binary Tree

Given an undirected planar graph G with n vertices and a set S of n points inside a simple polygon P, a point-set embedding of G on S is a planar drawing of G such that each vertex is mapped to a distinct point of S and the edges are polygonal chains surrounded by P. A special case of the embedding problem is that in which G is a balanced binary tree. In this paper, we present a new algorithm for embedding an n-vertex balanced binary tree BBT on a set S of n points inside a simple m-gon P in O(m4/3+ϵ + nlog2 n + m log n) time with at most O(m) bends per edge.

A randomized algorithm for finding a maximum clique in the visibility graph of a simple polygon

Discrete Algorithms We present a randomized algorithm to compute a clique of maximum size in the visibility graph G of the vertices of a simple polygon P. The input of the problem consists of the visibility graph G, a Hamiltonian cycle describing the boundary of P, and a parameter δ∈(0,1) controlling the probability of error of the algorithm. The algorithm does not require the coordinates of the vertices of P. With probability at least 1-δ the algorithm runs in O( |E(G)|2 / ω(G) log(1/δ)) time and returns a maximum clique, where ω(G) is the number of vertices in a maximum clique in G. A deterministic variant of the algorithm takes O(|E(G)|2) time and always outputs a maximum size clique. This compares well to the best previous algorithm by Ghosh et al. (2007) for the problem, which is deterministic and runs in O(|V(G)|2 |E(G)|) time.

Reprint of: Weighted straight skeletons in the plane

We investigate weighted straight skeletons from a geometric, graph-theoretical, and combinatorial point of view. We start with a thorough definition and shed light on some ambiguity issues in the procedural definition. We investigate the geometry, combinatorics, and topology of faces and the roof model, and we discuss in which cases a weighted straight skeleton is connected. Finally, we show that the weighted straight skeleton of even a simple polygon may be non-planar and may contain cycles, and we discuss under which restrictions on the weights and/or the input polygon the weighted straight skeleton still behaves similar to its unweighted counterpart. In particular, we obtain a non-procedural description and a linear-time construction algorithm for the straight skeleton of strictly convex polygons with arbitrary weights.

A Gradient Search Algorithm for the Maximal Visible Area Polygon Problem

This paper provides a gradient search algorithm for finding the maximal visible area polygon (VAP) viewed by an interior point in a simple polygon P. The algorithm is based on a natural partition of P into convex sets, such that each element of the partition is associated with a unique analytical form of the area function. We call this partition a back diagonal partition of P. Our maximal VAP algorithm converges in a finite number of steps, and is polynomial with a complexity of , for a simple polygon P with n vertices, and r reflex vertices. We use the maximal VAP algorithm as a basis for a greedy heuristic for the well known guardhouse problem with a computation complexity of .

GEODESIC-PRESERVING POLYGON SIMPLIFICATION

Polygons are a paramount data structure in computational geometry. While the complexity of many algorithms on simple polygons or polygons with holes depends on the size of the input polygon, the intrinsic complexity of the problems these algorithms solve is often related to the reflex vertices of the polygon. In this paper, we give an easy-to-describe linear-time method to replace an input polygon [Formula: see text] by a polygon [Formula: see text] such that (1) [Formula: see text] contains [Formula: see text], (2) [Formula: see text] has its reflex vertices at the same positions as [Formula: see text], and (3) the number of vertices of [Formula: see text] is linear in the number of reflex vertices. Since the solutions of numerous problems on polygons (including shortest paths, geodesic hulls, separating point sets, and Voronoi diagrams) are equivalent for both [Formula: see text] and [Formula: see text], our algorithm can be used as a preprocessing step for several algorithms and makes their running time dependent on the number of reflex vertices rather than on the size of [Formula: see text]. We describe several of these applications (including linear-time post-processing steps that might be necessary).

A fully automatic polygon scaled boundary finite element method for modelling crack propagation

An automatic crack propagation remeshing procedure using the polygon scaled boundary FEM is presented. The remeshing algorithm, developed to model any arbitrary shape, is simple yet flexible because only minimal changes are made to the global mesh in each step. Fewer polygon elements are used to predict the final crack path with the algorithm as compared to previous approaches. Two simple polygon optimisation methods which enable the remeshing procedure to model crack propagation more stably are implemented. Four crack propagation benchmarks are modelled to validate the developed method and demonstrate its salient features.