Simple deformation measures for discrete elastic rods and ribbons

The discrete elastic rod method (Bergou et al. 2008 ACM Trans. Graph . 27 , 63:1–63:12. ( doi:10.1145/1360612.1360662 )) is a numerical method for simulating slender elastic bodies. It works by representing the centreline as a polygonal chain, attaching two perpendicular directors to each segment and defining discrete stretching, bending and twisting deformation measures and a discrete strain energy. Here, we investigate an alternative formulation of this model based on a simpler definition of the discrete deformation measures. Both formulations are equally consistent with the continuous rod model. Simple formulae for the first and second gradients of the discrete deformation measures are derived, making it easy to calculate the Hessian of the discrete strain energy. A few numerical illustrations are given. The approach is also extended to inextensible ribbons described by the Wunderlich model, and both the developability constraint and the dependence of the energy on the strain gradients are handled naturally.

Evaluation of Two-Dimensional Angular Orientation of a Mobile Robot by a Modified Algorithm Based on Hough Transform

This paper proposes an algorithm that assesses the angular orientation of a mobile robot with respect to its referential position or a map of the surrounding space. In the framework of the suggested method, the orientation problem is converted to evaluating a dimensional rotation of the object that is abstracted as a polygon (or a closed polygonal chain). The method is based on Hough transform, which transforms the measurement space to a parametric space (in this case, a two-dimensional space [θ, r] of straight-line parameters). The Hough transform preserves the angles between the straight lines during rotation, translation, and isotropic scaling transformations. The problem of rotation assessment then becomes a one-dimensional optimization problem. The suggested algorithm inherits the Hough method’s robustness to noise.

Area collapse algorithm computing new curve of 2D geometric objects

The processing of cartographic data demands human involvement. Up-to-date algorithms try to automate a part of this process. The goal is to obtain a digital model, or additional information about shape and topology of input geometric objects. A topological skeleton is one of the most important tools in the branch of science called shape analysis. It represents topological and geometrical characteristics of input data. Its plot depends on using algorithms such as medial axis, skeletonization, erosion, thinning, area collapse and many others. Area collapse, also known as dimension change, replaces input data with lower-dimensional geometric objects like, for example, a polygon with a polygonal chain, a line segment with a point. The goal of this paper is to introduce a new algorithm for the automatic calculation of polygonal chains representing a 2D polygon. The output is entirely contained within the area of the input polygon, and it has a linear plot without branches. The computational process is automatic and repeatable. The requirements of input data are discussed. The author analyzes results based on the method of computing ends of output polygonal chains. Additional methods to improve results are explored. The algorithm was tested on real-world cartographic data received from BDOT/GESUT databases, and on point clouds from laser scanning. An implementation for computing hatching of embankment is described.

On Covering Points with Minimum Turns

We study the problem of finding a polygonal chain of line segments to cover a set of points in ℝd, d≥2, with the goal of minimizing the number of links or turns in the chain. A chain of line segments that covers all points in the given set is called a covering tour if the chain is closed, and is called a covering path if the chain is open. A covering tour or a covering path is rectilinear if all segments in the chain are axis-parallel. We prove that the two problems Minimum-Link Rectilinear Covering Tour and Minimum-Link Rectilinear Covering Path are both NP-hard in ℝ10.

Energy transfer within self-assembled cyclic multichromophoric arrays based on orthogonally arranged donor–acceptor building blocks

We herein present the coordination-driven supramolecular synthesis and photophysics of a [4+4] and a [2+2] assembly, built up by alternately collocated donor–acceptor chromophoric building blocks based, respectively, on the boron dipyrromethane (Bodipy) and perylene bisimide dye (PBI). In these multichromophoric scaffolds, the intensely absorbing/emitting dipoles of the Bodipy subunit are, by construction, cyclically arranged at the corners and aligned perpendicular to the plane formed by the closed polygonal chain comprising the PBI units. Steady-state and fs time-resolved spectroscopy reveal the presence of efficient energy transfer from the vertices (Bodipys) to the edges (PBIs) of the polygons. Fast excitation energy hopping – leading to a rapid excited state equilibrium among the low energy perylene-bisimide chromophores – is revealed by fluorescence anisotropy decays. The dynamics of electronic excitation energy hopping between the PBI subunits was approximated on the basis of a theoretical model within the framework of Förster energy transfer theory. All energy-transfer processes are quantitatively describable with Förster theory. The influence of structural deformations and orientational fluctuations of the dipoles in certain kinetic schemes is discussed.

Geodesic survey and modernization of a route as the task of optimization

A geodesic survey of an existing route requires one to determine the approximation curve by means of optimization using the total least squares method (TLSM). The objective function of the LSM was found to be a square of the Mahalanobis distance in the adjustment field ν. In approximation tasks, the Mahalanobis distance is the distance from a survey point to the desired curve. In the case of linear regression, this distance is codirectional with a coordinate axis; in orthogonal regression, it is codirectional with the normal line to the curve. Accepting the Mahalanobis distance from the survey point as a quasi-observation allows us to conduct adjustment using a numerically exact parametric procedure. Analysis of the potential application of splines under the NURBS (non-uniform rational B-spline) industrial standard with respect to route approximation has identified two issues: a lack of the value of the localizing parameter for a given survey point and the use of vector parameters that define the shape of the curve. The value of the localizing parameter was determined by projecting the survey point onto the curve. This projection, together with the aforementioned Mahalanobis distance, splits the position vector of the curve into two orthogonal constituents within the local coordinate system of the curve. A similar system corresponds to points that form the control polygonal chain and allows us to find their position with the help of a scalar variable that determines the shape of the curve by moving a knot toward the normal line.