Jacobi fields in nonholonomic mechanics

In this paper, we define Jacobi fields for nonholonomic mechanics using a similar characterization than in Riemannian geometry. We give explicit conditions to find Jacobi fields and finally we find the nonholonomic Jacobi fields in two equivalent ways: the first one, using an appropriate complete lift of the nonholonomic system and, in the second one, using the curvature and torsion of the associated nonholonomic connection.

Automated landmarking of bends in vascular structures: a comparative study with application to the internal carotid artery

Automated tools for landmarking the internal carotid artery (ICA) bends have the potential for efficient and objective medical image-based morphometric analysis. The two existing algorithms rely on numerical approximations of curvature and torsion of the centerline. However, input parameters, original source code, comparability, and robustness of the algorithms remain unknown. To address the former two, we have re-implemented the algorithms, followed by sensitivity analyses. Of the input parameters, the centerline smoothing had the least impact resulting in 6–7 bends, which is anatomically realistic. In contrast, centerline resolution showed to completely over- and underestimated the number of bends varying from 3 to 33. Applying the algorithms to the same cohort revealed a variability that makes comparison of results between previous studies questionable. Assessment of robustness revealed how one algorithm is vulnerable to model smoothness and noise, but conceptually independent of application. In contrast, the other algorithm is robust and consistent, but with limited general applicability. In conclusion, both algorithms are equally valid albeit they produce vastly different results. We have provided a well-documented open-source implementation of the algorithms. Finally, we have successfully performed this study on the ICA, but application to other vascular regions should be performed with caution.

Shape-changing chains for morphometric analysis of 2D and 3D, open or closed outlines

Morphometrics is a multivariate technique for shape analysis widely employed in biological, medical, and paleoanthropological applications. Commonly used morphometric methods require analyzing a huge amount of variables for problems involving a large number of specimens or complex shapes. Moreover, the analysis results are sometimes difficult to interpret and assess. This paper presents a methodology to synthesize a shape-changing chain for 2D or 3D curve fitting and to employ the chain parameters in stepwise discriminant analysis (DA). The shape-changing chain is comprised of three types of segments, including rigid segments that have fixed length and shape, scalable segments with a fixed shape, and extendible segments with constant curvature and torsion. Three examples are presented, including 2D mandible profiles of fossil hominin, 2D leaf outlines, and 3D suture curves on infant skulls. The results demonstrate that the shape-changing chain has several advantages over common morphometric methods. Specifically, it can be applied to a wide range of 2D or 3D profiles, including open or closed curves, and smooth or serrated curves. Additionally, the segmentation of profiles is a flexible and automatic protocol that can consider both biological and geometric features, the number of variables obtained from the fitting results for statistical analysis is modest, and the chain parameters that characterize the profiles can have physical meaning.

COMPUTATIONAL DESCRIPTION OF THE GEOMETRY OF ALIGNED CARBON NANOTUBES IN POLYMER NANOCOMPOSITES

The paper considers nanocomposites, reinforced with aligned carbon nanotubes (A- CNTs). Nominally aligned, the CNTs in the forest are wavy, which has important consequences in downgraded mechanical properties, and influences electric and thermal performance. The most detailed geometrical model of A-CNTs was proposed by Stein and Wardle (Nanotechnology, 27:035701, 2015). It creates a centerline trajectory of a CNT in steps, each step defining a section of the CNT, growing in the alignment direction with certain deviations. The paper, starting from this framework, formulates a model of the CNT geometry, which is based on the concept of correlation length of the CNT waviness and maximum admissible CNT curvature and torsion. The value of the maximum curvature can be linked to the buckling criteria for CNTs, or derived from ab initio and finite element modelling. It is used as a limiting factor for the growth, defining the waviness and tortuosity of the CNTs. The CNTs in the forest are placed in a random non-regular way, using Voronoi tessellation. The full paper includes investigation of the proposed algorithm for several values of the CNT volume fraction (in the range 0.5%…8%), the dependency of the modelled geometry on the curvature, and the apparent twist of the CNT centerlines. The modelling results are compared with experimental observations in 3D TEM imaging.

Decomposition of Velocity Field Along a Centerline Curve Using Frenet-Frames: Application to Arterial Blood Flow Simulations

This paper presents a novel method for the evaluation of three-dimensional blood-flow simulations based, on the decomposition of the velocity field into localized coordinate systems along the vessels centerline. The method is based on the computation of accurate centerlines with the Vascular Modeling Toolkit (VMTK) library, to calculate the localized Frenet-frames along the centerline and the morphological features, namely the curvature and torsion. Using the Frenet-frame unit vectors, the velocity field can be decomposed into axial, circumferential and radial components and visualized in a diagram along the centerline. This paper includes case studies with four idealized geometries resembling the carotid siphon and two patient-specific cases to demonstrate the capability of the method and the connection between morphology and flow. The proposed evaluation method presented in this paper can be easily extended to other derived quantities of the velocity fields, such as the wall shear stress field. Furthermore, it can be used in other fields of engineering with tubular cross-sections with complex torsion and curvature distribution.

Fitting Splines to Axonal Arbors Quantifies Relationship Between Branch Order and Geometry

Neuromorphology is crucial to identifying neuronal subtypes and understanding learning. It is also implicated in neurological disease. However, standard morphological analysis focuses on macroscopic features such as branching frequency and connectivity between regions, and often neglects the internal geometry of neurons. In this work, we treat neuron trace points as a sampling of differentiable curves and fit them with a set of branching B-splines. We designed our representation with the Frenet-Serret formulas from differential geometry in mind. The Frenet-Serret formulas completely characterize smooth curves, and involve two parameters, curvature and torsion. Our representation makes it possible to compute these parameters from neuron traces in closed form. These parameters are defined continuously along the curve, in contrast to other parameters like tortuosity which depend on start and end points. We applied our method to a dataset of cortical projection neurons traced in two mouse brains, and found that the parameters are distributed differently between primary, collateral, and terminal axon branches, thus quantifying geometric differences between different components of an axonal arbor. The results agreed in both brains, further validating our representation. The code used in this work can be readily applied to neuron traces in SWC format and is available in our open-source Python package brainlit: http://brainlit.neurodata.io/.

Classification of Holomorphic Functions as Pólya Vector Fields via Differential Geometry

We review Pólya vector fields associated to holomorphic functions as an important pedagogical tool for making the complex integral understandable to the students, briefly mentioning its use in other dimensions. Techniques of differential geometry are then used to refine the study of holomorphic functions from a metric (Riemannian), affine differential or differential viewpoint. We prove that the only nontrivial holomorphic functions, whose Pólya vector field is torse-forming in the cannonical geometry of the plane, are the special Möbius transformations of the form f(z)=b(z+d)−1. We define and characterize several types of affine connections, related to the parallelism of Pólya vector fields. We suggest a program for the classification of holomorphic functions, via these connections, based on the various indices of nullity of their curvature and torsion tensor fields.

Representation of the kinematics of the natural trihedral of a spiral-helix trajectory by quaternion matrices

The spiral-helix trajectory of the transport vehicle programmed motion in the form of a hodograph in the stationary frame of reference is considered. A relative frame of reference associated with the natural trihedral of the trajectory is introduced. The formulas of curvature and torsion of the trajectory, the unit vector of the natural trihedral, the components of the angular velocity of rotation of the natural trihedral in the proper axes and in the stationary frame of reference are set in the quaternionic matrices. The results are verified using the Frenet-Serret formulas. The mathematical apparatus of quaternion matrices is tested with the aim of adapting spatial, nonlinear problems of dynamic design of transport vehicles to a computational experiment.

CALCULATION MODEL OF A COMPLEX STRESS REINFORCED CONCRETE ELEMENT OF A BOXED SECTION DURING TORSION WITH BENDING

Statement of the problem. Based on the analysis of domestic and foreign scientific publications and guidelines, it is found that the known deformation models for the calculation of complex tensile reinforced concrete elements during torsional bending are quite conditional. Therefore the article considers the solution of the problem of designing a computational model of a reinforced concrete element during torsion with bending in the post-crack stage, which most fully accounts for the specifics of crack formation, deformation and destruction of such elements. The case is considered for when among all possible external influences the action of torques and bending moments has the greatest influence on the stress-strain. Results. Using the equations of statics and physical ratios of reinforced concrete, the calculated parameters are identified such as stresses in concrete of compressed zone, height of compressed concrete, stresses in clamps, deformations in concrete and reinforcement, curvature and torsion angle of reinforced concrete element. Conclusions. The obtained analytical dependences were tested by means of numerical calculation of the reinforced concrete strapping crossbar of the outer contour of a residential building of box section of high-strength concrete. The suggested deformation model can be employed in the design of a wide class of reinforced concrete structures working on torsional bending.

INVOLUTE-EVOLUTE CURVES ACCORDING TO MODIFIED ORTHOGONAL FRAME

In this paper, the involute-evolute curve concept has been defined according to two type modified orthogonal frames at non-zero points of curvature and torsion in the Euclidean space E^3 , respectively. Later, the characteristic theorems related to the distance between the corresponding points of these curves have been given. Besides, the relations have been found between the curvatures and also torsions of the two type the involute-evolute modified orthogonal pairs.