On vertex PI index of certain triangular tessellation networks

The Wiener index, due to its many applications is considered to be one of very important distance-based index. But the Padmaker-Ivan (PI) index is kind of the only distance related index linked to parallelism of edges. The PI index like other distance related indices has great disseminating power. The index was firstly investigated by Khadikar et al. (2001), they have probed the chemical applications of the PI index. They proved that the proposed PI index correlates highly with the physicochemical properties and biological activities of a large number of diverse and complex chemical compounds and the Wiener and Szeged indices. Recently, the vertex Padmarkar-Ivan (PIv) index of a chemical graph G was introduced as the sum over all edges uv of a molecular graph G of the vertices of the graph that are not equidistant to the vertices u and v. In this paper, the vertex PIv index of certain triangular tessellation are computed by using graph-theoretic analysis, combinatorial computing, and edge-dividing technology.

Bijective, Non-Bijective and Semi-Bijective Translations on the Triangular Plane

The triangular plane is the plane which is tiled by the regular triangular tessellation. The underlying discrete structure, the triangular grid, is not a point lattice. There are two types of triangle pixels. Their midpoints are assigned to them. By having a real-valued translation of the plane, the midpoints of the triangles may not be mapped to midpoints. This is the same also on the traditional square grid. However, the redigitized result on the square grid always gives a bijection (gridpoints of the square grid are mapped to gridpoints in a bijective way). This property does not necessarily hold on to the triangular plane, i.e., the redigitized translated points may not be mapped to the original points by a bijection. In this paper, we characterize the translation vectors that cause non bijective translations. Moreover, even if a translation by a vector results in a bijection after redigitization, the neighbor pixels of the original pixels may not be mapped to the neighbors of the resulting pixel, i.e., a bijective translation may not be digitally ‘continuous’. We call that type of translation semi-bijective. They are actually bijective but do not keep the neighborhood structure, and therefore, they seemingly destroy the original shape. We call translations strongly bijective if they are bijective and also the neighborhood structure is kept. Characterizations of semi- and strongly bijective translations are also given.

Clearance-driven motion planning for mobile robots with differential constraints

SUMMARYThis paper presents an approach that integrates the geometric notion of clearance (distance to the closest obstacle) into sampling-based motion planning to enable a robot to safely navigate in challenging environments. To reach the goal destination, the robot must obey geometric and differential constraints that arise from the underlying motion dynamics and the characteristics of the environment. To produce safe paths, the proposed approach expands a motion tree of collision-free and dynamically feasible motions while maintaining locally maximal clearance. In distinction from related work, rather than explicitly constructing the medial axis, the proposed approach imposes a grid or a triangular tessellation over the free space and uses the clearance information to construct a weighted graph where edges that connect regions with low clearance have high cost. Minimum-cost paths over this graph produce high-clearance routes that tend to follow the medial axis without requiring its explicit construction. A key aspect of the proposed approach is a route-following component which efficiently expands the motion tree to closely follow such high-clearance routes. When expansion along the current route becomes difficult, edges in the tessellation are penalized in order to promote motion-tree expansions along alternative high-clearance routes to the goal. Experiments using vehicle models with second-order dynamics demonstrate that the robot is able to successfully navigate in complex environments. Comparisons to the state-of-the-art show computational speedups of one or more orders of magnitude.

Positional Uncertainty Estimation and Visualization of a Triangular Tessellation

A triangulated irregularly network (TIN) is always used to describe the land surface in a geographic information system (GIS), while the land surface abstracted and simplified has modeling errors inevitably, which play an important role in uncertainty estimation, while are ignored in the GIS literature. In this paper, we consider modeling errors and use error promulgation law to estimate positional uncertainty of a triangular tessellation. It indicates the difference between complex entities in the real world and geographic objects simplified in a digital world, and it makes uncertainty complete and credible.

Computing Divergence from a Surface Network: Comparison of the Triangle and Pentagon Methods

Two methods for creating gridded fields of divergence from irregularly spaced wind observations are evaluated by sampling analytic fields of cyclones and anticyclones of varying wavelengths using a surface network. For the triangle method, which requires a triangular tessellation of the station network and assumes that the wind varies linearly within each triangle, divergence estimates are obtained directly from the wind observations and are assumed valid at triangle centroids. These irregularly spaced centroid divergence estimates then are analyzed to a grid using a Barnes analysis scheme. For the pentagon method, which requires a pentagonal tessellation of the station network and assumes that the wind varies quadratically within each pentagon, divergence estimates also are obtained directly from the wind observations and are valid at the station lying within the interior of each pentagon. These irregularly spaced divergence estimates then are analyzed to a grid using the same Barnes analysis scheme. It is found that for errorless observations, the triangle method provides better analyses than the pentagon method for all wavelengths considered, despite the more restrictive assumption by the triangle method regarding the wind field. For well-sampled wavelengths, however, the preanalyzed divergence estimates at the interior stations of pentagons are found to be superior to those at triangle centroids. When random, Gaussian errors are added to the observations, all advantages of the pentagon method over the triangle method are found to disappear.