An exact algorithm for minimizing vertex guards on art galleries

We consider the problem [art gallery problem (AGP)] of minimizing the number of vertex guards required to monitor an art gallery whose boundary is an n‐vertex simple polygon. In this paper, we compile and extend our research on exact approaches for solving the AGP. In prior works, we proposed and tested an exact algorithm for the case of orthogonal polygons. In that algorithm, a discretization that approximates the polygon is used to formulate an instance of the set cover problem, which is subsequently solved to optimality. Either the set of guards that characterizes this solution solves the original instance of the AGP, and the algorithm halts, or the discretization is refined and a new iteration begins. This procedure always converges to an optimal solution of the AGP and, moreover, the number of iterations executed highly depends on the way we discretize the polygon. Notwithstanding that the best known theoretical bound for convergence is Θ(n³) iterations, our experiments show that an optimal solution is always found within a small number of them, even for random polygons of many hundreds of vertices. Herein, we broaden the family of polygon classes to which the algorithm is applied by including non‐orthogonal polygons. Furthermore, we propose new discretization strategies leading to additional trade‐off analysis of preprocessing vs. processing times and achieving, in the case of the novel Convex Vertices strategy, the most efficient overall performance so far. We report on experiments with both simple and orthogonal polygons of up to 2500 vertices showing that, in all cases, no more than 15 minutes are needed to reach an exact solution, on a standard desktop computer. Ultimately, we more than doubled the size of the largest instances solved to optimality compared with our previous experiments, which were already five times larger than those previously reported in the literature.     

Robust and Optimized Algorithms for the Point‐in‐Polygon Inclusion Test without Pre‐processing

In this work we present new point inclusion algorithms for non‐convex polygons. These algorithms do not perform any pre‐processing or any type of decomposition nor features classification, which makes them especially suitable for deformable or moving polygons. The algorithms are more accurate and robust than others in the sense that they consider the inclusion of the point in the vertices and edges of the polygon, and deal with the special cases correctly. In order to perform this inclusion test efficiently, they use the sign of the barycentric coordinates of the test point with regard to the triangles formed by the edges and an origin that depends on the test point. This set of triangles, which is a special simplicial covering of the polygon, is constructed after a transformation of the polygon that simplifies the calculations involved in the inclusion test. Then, an appropriate ordering of the rejection tests allows us to optimize this method. Our algorithms have been tested for robustness and compared with ray‐crossing methods, showing a significant improvement.     

求解简单多边形间最小距离的一个线性时间算法

计算简单多边形间的最小距离,在所有与几何图形计算有关的领域中,一直以来都是一个基本问题。为了更快地求解简单多边形的最小距离,提出了一个基于关联多边形三角化分割的简单多边形间最小距离的求解算法。该算法的主要思想是:首先构造一个关联多边形把两个多边形联系起来,其目的是把最小距离限制在这个关联多边形内;然后根据两个多边形的最小边界矩形包围框间的不同位置关系,详细阐述了关联多边形的构造过程,同时论述了关联多边形是一个简单多边形。为了计算最小距离,首先要对关联多边形进行三角化分割,并使最小距离位于三角化分割结果中某一个三角形区域内,或者至多位于两个相邻三角形区域内;之后通过对所有三角形进行遍历来找出最小距离及其所在的位置。该算法的时间复杂度是线性的。     

一种加权剖分简单多边形为三角形和凸四边形子域的算法

针对计算几何与有限元网格自动剖分中多边形子域剖分问题，给出了一种适用于有限元网格子域单元（即大单元）剖分的标准，并提出了一种通过在可视点对之间引进适当的多边形剖分和根据子域单元的形状质量判定因子来引导剖分的算法。由于建立的权函数和凹角（凸角）本身有关，因此对同属于凹角（凸角）的权函数也可以加以权值上的区分。该算法通过分步进行剖分，即先将简单多边形剖分为凸多边形，然后再将凸多边形剖分为凸六边形和凸五边形，最后将凸六边形和凸五边形剖分为三角形和凸四边形，以得到满足要求的剖分结果。在以上的每个剖分过程中，都引进了权重来引导剖分，使得剖分结果更加优化、合理。     

Optimality and competitiveness of exploring polygons by mobile robots

A mobile robot, represented by a point moving along a polygonal line in the plane, has to explore an unknown polygon and return to the starting point. The robot has a sensing area which can be a circle or a square centered at the robot. This area shifts while the robot moves inside the polygon, and at each point of its trajectory the robot “sees” (explores) all points for which the segment between the robot and the point is contained in the polygon and in the sensing area. We focus on two tasks: exploring the entire polygon and exploring only its boundary. We consider several scenarios: both shapes of the sensing area and the Manhattan and the Euclidean metrics.We focus on two quality benchmarks for exploration performance: optimality (the length of the trajectory of the robot is equal to that of the optimal robot knowing the polygon) and competitiveness (the length of the trajectory of the robot is at most a constant multiple of that of the optimal robot knowing the polygon). Most of our results concern rectilinear polygons. We show that optimal exploration is possible in only one scenario, that of exploring the boundary by a robot with square sensing area, starting at the boundary and using the Manhattan metric. For this case we give an optimal exploration algorithm, and in all other scenarios we prove impossibility of optimal exploration. For competitiveness the situation is more optimistic: we show a competitive exploration algorithm for rectilinear polygons whenever the sensing area is a square, for both tasks, regardless of the metric and of the starting point. Finally, we show a competitive exploration algorithm for arbitrary convex polygons, for both shapes of the sensing area, regardless of the metric and of the starting point.     

对体可视化Marching Cube算法的改进

提出一个由三维数据计算等值面中点的算法,它在两方面对标准Ｍａｒｃｈｉｎｇ Ｃｕｂｅ算法进行了改进。第一个改进是：在等值面上样本点的状态依赖于它所连接的边同等值面相交的数目;第二个改进是：两相邻样本点中等值面多边形顶点被定位于中点,使得共面三角片合并为一个多边莆,减少了生成多边形的数量,提高了算法效率。     

Solving the irregular strip packing problem via guided local search for overlap minimization

The irregular strip-packing problem (ISP) requires a given set of non-convex polygons to be placed without overlap within a rectangular container having a fixed width and a variable length, which is to be minimized. As a core sub-problem to solve ISP, we consider an overlap minimization problem (OMP) whose objective is to place all polygons into a container with given width and length so that the total amount of overlap between polygons is made as small as possible. We propose to use directional penetration depths to measure the amount of overlap between a pair of polygons, and present an efficient algorithm to find a position with the minimum overlap for each polygon when it is translated in a specified direction. Based on this, we develop a local search algorithm for OMP that translates a polygon in horizontal and vertical directions alternately. Then we incorporate it in our algorithm for OMP, which is a variant of the guided local search algorithm. Computational results show that our algorithm improves the best-known values of some well-known benchmark instances.     

计算多边形交集、并集面积的算法

平面多边形交集与并集面积的计算机算法可以利用多边形裁剪算法来实现。本文提出的算法思想是利用Weiler-Atherton多边形裁剪算法中的多边形链表,在遍历链表时遇到交点就改变跟踪方向,这样可以求出并集顶点表,求交集时只要从入点开始跟踪遇到交点再改变跟踪方向;最后,通过交集和并集表求出它们的面积。多边形可以是凸的或凹的、甚至是带孔的。     

kDet: Parallel Constant Time Collision Detection for Polygonal Objects

We define a novel geometric predicate and a class of objects that enables us to prove a linear bound on the number of intersecting polygon pairs for colliding 3D objects in that class. Our predicate is relevant both in theory and in practice: it is easy to check and it needs to consider only the geometric properties of the individual objects – it does not depend on the configuration of a given pair of objects. In addition, it characterizes a practically relevant class of objects: we checked our predicate on a large database of real‐world 3D objects and the results show that it holds for all but the most pathological ones. Our proof is constructive in that it is the basis for a novel collision detection algorithm that realizes this linear complexity also in practice. Additionally, we present a parallelization of this algorithm with a worst‐case running time that is independent of the number of polygons. Our algorithm is very well suited not only for rigid but also for deformable and even topology‐changing objects, because it does not require any complex data structures or pre‐processing. We have implemented our algorithm on the GPU and the results show that it is able to find in real‐time all colliding polygons for pairs of deformable objects consisting of more than 200k triangles, including self‐collisions.     

基于三角形分解和重构的平面多边形变形方法

为解决较复杂的不同拓扑结构的二维形状渐变问题,提出一种基于三角形分解和重构的平面多边形变形方法.该方法将图形多层分解为三角形,保留分解过程中的各层边角信息;然后通过线性插值各层边长比例及角度,并结合刚性变换方法重构中间多边形的细节和框架,以达到变形的目的.该方法适用于任意点数的多边形,具有一般性.实验结果表明,文中方法能很好地解决变形序列中的萎缩问题,并且对较复杂的狭长图形也能避免自交现象,变形效果自然.     

Barycentric Coordinates on Surfaces

This paper introduces a method for defining and efficiently computing barycentric coordinates with respect to polygons on general surfaces. Our construction is geared towards injective polygons (polygons that can be enclosed in a metric ball of an appropriate size) and is based on replacing the linear precision property of planar coordinates by a requirement in terms of center of mass, and generalizing this requirement to the surface setting. We show that the resulting surface barycentric coordinates can be computed using planar barycentric coordinates with respect to a polygon in the tangent plane. We prove theoretically that the surface coordinates properly generalize the planar coordinates and carry some of their useful properties such as unique reconstruction of a point given its coordinates, uniqueness for triangles, edge linearity, similarity invariance, and smoothness; in addition, these coordinates are insensitive to isometric deformations and can be used to reconstruct isometries. We show empirically that surface coordinates are shape‐aware with consistent gross behavior across different surfaces, are well‐behaved for different polygon types/locations on variety of surface forms, and that they are fast to compute. Finally, we demonstrate effectiveness of surface coordinates for interpolation, decal mapping, and correspondence refinement.     

三维网格模型的布尔运算方法

提出了一种基于三维网格模型的布尔运算方法。首先通过基于方向包围盒（OBB）层次包围盒树的碰撞检测算法,得到实体的相交三角形对;接下来求出两相交三角形之间的交线,建立与三角形的交线拓扑关系;通过分类处理三种交线类型来对相交三角形进行区域划分,得到一系列多边形,并对多边形进行三角剖分形成结果区域;最后根据体的包含关系构建关系邻接表,判断多边形区域的相对于其他实体的内外关系并通过网格模型的拓扑关系,定位表面三角网格区域;同时根据交、并、差等布尔操作,对结果区域进行取舍,得到最终结果。实验结果表明相交部分的岩性与实体的岩性相吻合,验证了该算法的正确性以及可行性。     

Shortest path between two simple polygons

Consider a collection of mutually disjoint simple polygons in the plane containing a total of n edges. Two of them are specified as a source polygon S and a target polygon T. We present an efficient algorithm for finding a shortest path between S and T avoiding the other polygons. We show that it runs in O(n²) time, using a linear-time algorithm for computing the visibility polygon of a point. This problem is related to a wire routing design of a certain type of LSI for which terminals are of polygonal shape and larger than a wire segment.     

基于Delaunay三角网的任意多边形三角剖分算法研究

文章通过分析现有多边形三角剖分算法,给出一种基于Delaunay三角网的任意复杂多边形三角剖分的改进算法。算法首先忽略多边形顶点与边线间的逻辑关系,将其看做散乱顶点的集合,然后采用Delaunay三角化方法对点集进行合理剖分,再依据多边形顶点及边线间的逻辑关系,逐一将那些不合理的三角网剔除,最终重新组合出符合要求的三角网格。     

A Fast Hidden-line Removal Algorithm for 3-Dimensional Buildings

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求解单位等边三角形Packing问题的近似算法

多边形Packing问题不仅具有重要的理论意义，而且也有广阔的应用前景，由于该问题具有NP难度，且具有连续的性质，一般要事先对多边形的放置方位进行限制，例如不允许多边形旋转，然后再进行优化求得近似解，该文采用一种新的思路对多边形Packing问题的一个特例－单位等边三角形Packing问题进行了研究，提出了零自由度动作和零自由度放置策略的概念，并设计了一个近似求解算法－最小损伤法，复杂性分析和计算结果表明该算法是高效的，以此为基础，可能为多边形Packing问题找到类似的求解算法。     

基于夹角变化趋势的多边形自动搜索和生成算法

利用左转算法生成多边形是GIS中面域组织和拓扑关系建立的常用算法。根据算法规则,对于由顺时针方向和逆时针方向建立的多边形都可以生成多边形文件,这就会产生一些重复多边形和无效的多边形。为此,提出了基于夹角变化趋势判断多边形搜索方向的算法,根据左转或右转算法得到的点组顺序,分别计算由起始点出发的弧段的方位角,根据相邻弧段夹角的和来判断多边形的搜索方向,实现了每一多边形都是由左转算法生成,完成了多边形的自动建立。该算法有效地判断了多边形的搜索方向,避免了无效多边形的生成。     

SDS模型化简合并多边形的一个改进算法研究

作为自动制图综合中的重要组成部分，多边形化简与合并可用来解决由于制图比例尺减小而带来的多边形与多边形之间以及多边形内部的邻近冲突问题。该方法设计主要基于多边形几何特征、拓扑特征的分析，同时需要一种合适的空间数据模型用于支持多边形与多边形之间，多边形内部的邻近关系探测。为避免分离操作带来的多边形综合结果的不同，在分析了基于Delaunay三角网的SDS模型表达对象邻近关系的能力后，提出了一种统一解决多边形化简和合并的方案，同时对原有的邻近冲突检测方法进行了改进，从而解决了由于边缘尖锐三角形的引入而导致多边形合并和化简后面积大量增大和丢失某些特征点的问题。实验结果表明，该方法可以获得满意的多边形化简合并效果。     

Approximate Guarding of Monotone and Rectilinear Polygons

We show that vertex guarding a monotone polygon is NP-hard and construct a constant factor approximation algorithm for interior guarding monotone polygons. Using this algorithm we obtain an approximation algorithm for interior guarding rectilinear polygons that has an approximation factor independent of the number of vertices of the polygon. If the size of the smallest interior guard cover is OPT for a rectilinear polygon, our algorithm produces a guard set of size O(OPT ²).     

一种有效的任意多边形裁剪算法

介绍了一种基于改进的Weiler算法的任意多边形裁剪算法，该算法通过引入图形部件和合理的数据结构来组织裁剪后的多边形，减少了遍历多边形顶点链表的次数，并有效减少求交点的时间，具有占用存储空间少和处理速度快的特点。经过实例测试，算法对同时处理单个和多个任意多边形裁剪具有良好的稳定性、可靠性和较高的效率。