Skeletons Of Polygons Research Articles

Rational design, structure properties, and synthesis strategies of dual-pore covalent organic frameworks (COFs) for potent applications: A review

Dual-pore covalent organic frameworks (COFs) offer a molecular scaffold for introducing building blocks into periodically organized polygonal skeletons to produce fascinating structural features. The rapid development of this material has attracted intensive interest from researchers with diverse expertise. This review selects the leading scientific findings about dual-pore COFs and highlights their functions and perspectives on design, structure properties, and synthesis strategies. Dual-pore COFs, as newly hetero-pore COFs by integrating particular pores into one polygonal skeleton, have been compared to conventional COFs. Dual-pore COFs display hierarchical/heterogeneous porosities and homogeneous porosity, which endow them with exceptional features involving mass diffusion, charge transfer, and large surface area with abundant active sites. Additionally, the strategic dual-pore design by opting for different approaches, such as integration of [D2h + C2] symmetries, kagome-type lattices, and other symmetric arrangements of monomers, are inclusively discussed. Identification and construction of dual-pores in COFs via optimal synthetic methods, such as desymmetrization, multiple linking sites, and orthogonal reactions, are highlighted as the primary pore engineering routes to simultaneously regulate the growth and alter the characteristics of COFs for promising applications. Lastly, a focused discussion on various challenges and critical fundamentals of dual-pore engineering is successfully outlined, with potential prospects of introducing dual-pore in COFs.

An exact algorithm for constructing minimum Euclidean skeletons of polygons

An exact algorithm for constructing minimum Euclidean skeletons of polygons

Mitered Offsets and Skeletons for Circular Arc Polygons

The offsetting process that defines straight skeletons of polygons is generalized to arc polygons, i.e., to planar shapes with piecewise circular boundaries. The offsets are obtained by shrinking or expanding the circular arcs on the boundary in a co-circular manner, and tracing the paths of their endpoints. These paths define the associated shape-preserving skeleton, which decomposes the input object into patches. While the skeleton forms a forest of trees, the patches of the decomposition have a radial monotonicity property. Analyzing the events that occur during the offsetting process is non-trivial; the boundary of the offsetting object may get into self-contact and may even splice. This leads us to an event-driven algorithm for offset and skeleton computation. Several examples (both manually created ones and approximations of planar free-form shapes by arc spline curves) are analyzed to study the practical performance of our algorithm.

Bio-inspired polygonal skeleton structure for vibration isolation: Design, modelling, and experiment

Inspired by the safe landing of a cat falling from a high altitude, a bio-inspired polygonal skeleton (BIPS) structure is proposed, and its nonlinear characteristics are systematically studied to explore its potential application in the suppression of vibration. The polygon is formed by the skeleton structure of the cat’s entire body and the ground. The BIPS system consists of two symmetrical bionic legs with three robs (as skeleton) and four horizontal springs (as muscle). Two bionic legs are connected through the bearing platform (as spine), which could adjust the distance between the two bionic legs. A theoretical model is developed to characterize its stiffness nonlinearity through geometrical and mechanical analysis. Parameter analysis reveals that the BIPS structure has diverse stiffness, including nonlinear positive stiffness and negative stiffness. By imitating adjustment of leg posture and telescopic function of the spine (control the distance between legs), these flexible stiffness properties can be adjusted by structure parameters. In addition, the load capacity and working range can also be designed by the length of the bars, the initial angle, the mounting position, and the spring stiffness. The experimental setup is established, and the vibration isolation performance under various excitation is tested. The experimental results verify the accuracy of the dynamic model and also show that the proposed BIPS structure can suppress the vibration effectively under a variety of excitations. These peculiarities may provide potential possibility of an innovative approach to passive vibration control and isolation.

Barycentric convolution surfaces based on general planar polygon skeletons

Using barycentric coordinates for thickness interpolation, we present a novel polygonal skeleton based convolution surface approximation method with varying radii. Given the prescribed radii of an arbitrary planar polygonal skeleton, we first employ a smooth interior mean value coordinate interpolation approach to calculate the thickness at each projection position in the polygonal plane. Then a local thickness approximation method based on finite-support kernels is introduced to create an implicit surface with smoothly varying thickness. In addition, a polygon offset with different distances by winding numbers is employed to create local approximation at polygon boundaries. Our experiments show that the proposed uniform smooth thickness interpolation and local convolution approximation method can not only avoid surface wrinkles but also reduce computation cost. Moreover, our approach is insensitive to exterior thickness interpolation. Therefore smooth barycentric coordinates within a polygon can all be easily integrated into our approach.

Min-/Max-Volume Roofs Induced by Bisector Graphs of Polygonal Footprints of Buildings

Piecewise-linear terrains (“roofs”) over simple polygons were first studied by Aichholzer et al. (J. UCS 1995) in their work on straight skeletons of polygons. We show how to construct a roof over the polygonal footprint of a building that has minimum or maximum volume among all roofs that drain water. Our algorithm for computing such a roof extends the standard plane-sweep approach known from the theory of straight skeletons by additional events. For both types of roofs our algorithm runs in [Formula: see text] time for a simple polygon with [Formula: see text] vertices.

Perceptually stable regions for arbitrary polygons

Zou and Yan have recently developed a skeletonization algorithm of digital shapes based on a regularity/singularity analysis; they use the polygon whose vertices are the boundary pixels of the image to compute a constrained Delaunay triangulation (CDT) in order to find local symmetries and stable regions. Their method has produced good results but it is slow since its complexity depends on the number of contour pixels. This paper presents an extension of their technique to handle arbitrary polygons, not only polygons of short edges. Consequently, not only can we achieve results as good as theirs for digital images, but we can also compute skeletons of polygons of any number of edges. Since we can handle polygonal approximations of figures, the skeletons are more resilient to noise and faster to process.

Deforming Geometric Models Based on a Polygonal Skeleton Mesh

A technique is presented for the deformation of a given geometric model (to be called: the skin) by means of a polygonized skeleton model. It is based on recognition of similarities in shape between tehse models by means of a method called dual-connectivity search. The technique uses these similarities in shape to define a semantic relation between the skeleton and the skin such that a smooth deformation for the skin is generated, that resembles, and is derived from, a deformation of the skeleton.

Upper and Lower Bounds for the Number of Conjugated Patterns in Carbocyclic and Heterocyclic Compounds

Abstract It is possible to regard two polygonal skeletons as the same in a special class of carbocyclic and heterocyclic compounds, if the one is reducible to the other by means of the contraction of cyclic subskeletons, and if the numbers of conjugated patterns in them are equal to each other. In such polygonal skeletons, three forms of cyclic subskeletons are defined; the one is called “alternate”, and the others, involving the one called “inclusive”, have a path (b, b), where (b) is a conjugated vertex connecting with three vertices. Successive eliminations of the cyclic subskeletons enable to estimate the upper and lower bounds for the number of conjugated patterns in a given polygonal skeleton.