Research on the Coverage of Ultraviolet Communication Network in the Arbitrary Polygon Area

Given that the current ultraviolet (UV) networking model is established in a regular circular area, this article studies the coverage of a UV non-line-of-sight (NLOS) communication network creatively in the arbitrary polygon area. In this paper, the UV communication model and the basic concepts of network coverage are introduced first. Then the influence parameters of the UV node communication radius are studied, and the changes of the communication radius under different work patterns are analyzed. Finally, the coverage of the square target area is simulated under different communication parameters (transmitted power, data rate and node density). The results illustrate that the smaller the transceiver elevation angles are, the better the network coverage performance is. Additionally, we numerically compare the UV network models of polygonal and circular regions, which can be used as a reference for actual networking.

Simultaneous Smoothing and Untangling of 2D Meshes Based on Explicit Element Geometric Transformation and Element Stitching

Mesh quality can affect both the accuracy and efficiency of numerical solutions. This paper first proposes a geometry-based smoothing and untangling method for 2D meshes based on explicit element geometric transformation and element stitching. A new explicit element geometric transformation (EEGT) operation for polygonal elements is firstly presented. The transformation, if applied iteratively to an arbitrary polygon (even inverted), will improve its regularity and quality. Then a well-designed element stitching scheme is introduced, which is achieved by carefully choosing appropriate element weights to average the temporary nodes obtained by the above individual element transformation. Based on the explicit element geometric transformation and element stitching, a new mesh smoothing and untangling approach for 2D meshes is proposed. The proper choice of averaging weights for element stitching ensures that the elements can be transitioned smoothly and uniformly throughout the calculation domain. Numerical results show that the proposed method is able to produce high-quality meshes with no inverted elements for highly tangled meshes. Besides, the inherent regularity and fine-grained parallelism make it suitable for implementation on Graphic Processor Unit (GPU).

A General Leaf Area Geometric Formula Exists for Plants—Evidence from the Simplified Gielis Equation

Plant leaves exhibit diverse shapes that enable them to utilize a light resource maximally. If there were a general parametric model that could be used to calculate leaf area for different leaf shapes, it would help to elucidate the adaptive evolutional link among plants with the same or similar leaf shapes. We propose a simplified version of the original Gielis equation (SGE), which was developed to describe a variety of object shapes ranging from a droplet to an arbitrary polygon. We used this equation to fit the leaf profiles of 53 species (among which, 48 bamboo plants, 5 woody plants, and 10 geographical populations of a woody plant), totaling 3310 leaves. A third parameter (namely, the floating ratio c in leaf length) was introduced to account for the case when the theoretical leaf length deviates from the observed leaf length. For most datasets, the estimates of c were greater than zero but less than 10%, indicating that the leaf length predicted by the SGE was usually smaller than the actual length. However, the predicted leaf areas approximated their actual values after considering the floating ratios in leaf length. For most datasets, the mean percent errors of leaf areas were lower than 6%, except for a pooled dataset with 42 bamboo species. For the elliptical, lanceolate, linear, obovate, and ovate shapes, although the SGE did not fit the leaf edge perfectly, after adjusting the parameter c, there were small deviations of the predicted leaf areas from the actual values. This illustrates that leaves with different shapes might have similar functional features for photosynthesis, since the leaf areas can be described by the same equation. The anisotropy expressed as a difference in leaf shape for some plants might be an adaptive response to enable them to adapt to different habitats.

Reconstruction of Weakly Simple Polygons from Their Edges

We address the problem of reconstructing a polygon from the multiset of its edges. Given [Formula: see text] line segments in the plane, find a polygon with [Formula: see text] vertices whose edges are these segments, or report that none exists. It is easy to solve the problem in [Formula: see text] time if we seek an arbitrary polygon or a simple polygon. We show that the problem is NP-complete for weakly simple polygons, that is, a polygon whose vertices can be perturbed by at most [Formula: see text], for any [Formula: see text], to obtain a simple polygon. We give [Formula: see text]-time algorithms for reconstructing weakly simple polygons: when all segments are collinear or the segment endpoints are in general position. These results extend to the variant in which the segments are directed. We study related problems for the case that the union of the [Formula: see text] input segments is connected. (i) If each segment can be subdivided into several segments, find the minimum number of subdivision points to form a weakly simple polygon. (ii) If new line segments can be added, find the minimum total length of new segments that creates a weakly simple polygon. We give worst-case upper and lower bounds for both problems.