Curvature Of Curve Research Articles

Generalized evolutes of planar curves

The evolutes of regular curves in the Euclidean plane are given by the caustics of regular curves. In this paper, we define the generalized evolutes of planar curves which are spatial curves, and the projection of generalized evolutes along a fixed direction are the evolutes. We also prove that the generalized evolutes are the locus of centers of slant circles of the curvature of planar curves. Moreover, we define the generalized parallels of planar curves and show that the singular points of generalized parallels sweep out the generalized evolute. In general, we cannot define the generalized evolutes at the singular points of planar curves, but we can define the generalized evolutes of fronts by using moving frames along fronts and curvatures of the Legendre immersion. Then we study the behaviors of generalized evolutes at the singular points of fronts. Finally, we give some examples to show the generalized evolutes.

Passive static stretching alters the characteristics of the force-velocity curvature differently for fast and slow muscle groups-A practical application of Hill's equation.

Studies showed fast muscle fibers have a greater constant b value of Hill's equation than that of slow muscle fibers, and the changing ratio of b/Vmax indicates the altered characteristics of muscles under certain conditions such as static stretching. This study was to investigate the effect of acute passive static stretching on the curvature of force-velocity curve in people with different muscle fiber types. A two-step work was conducted in current study through using Hill's equation: 1) calculated b values for each subject at different conditions (non-stretched and stretched) to determine muscle groups, and 2) examined the effect of static stretching on different muscle groups. Sixty-five college students performed isokinetic leg extensions at 5 speeds to test peak torque, following either a non-stretching or two passive static quadriceps stretching exercises. The peak torque and corresponding velocity were used to calculate the b constant. Data reduction consisted of calculating a Z score for each non-stretched and stretched b values. Individuals, whose non-stretched b constant was above or below one standard deviation of the Z score, were designated as the less curved (fast) and more curved (slow) groups, respectively. A paired t-test was used to analyze the pre and post intervention effect on b values for each group (p<0.05). This study found passive static stretching significantly altered the b constant of the fast group, but no effect on slow group. Therefore, we suggest static stretching should be avoided immediately before fast or explosive activities in individuals using predominantly fast muscle fibers.

Existence of curves with constant geodesic curvature in a Riemannian 2-sphere

We prove the existence of immersed closed curves of constant geodesic curvature in an arbitrary Riemannian 2-sphere for almost every prescribed curvature. To achieve this, we develop a min-max scheme for a weighted length functional.

Testing and numerical modelling of circular CFDST cross-sections with stainless steel outer tubes in bending

The structural performance of circular concrete-filled double skin tubular (CFDST) cross-sections with stainless steel outer tubes has been examined herein based on experiments and numerical modelling. A laboratory testing programme comprising a total of 22 four-point bending tests was performed on seven CFDST cross-sections with varying concrete grades. The details of the test rig and procedures, as well as the key test observations, including the failure moment capacities, moment–curvature curves and failure modes, are fully reported. A numerical modelling programme was then carried out; a finite element (FE) model was first established to replicate the test observations, and then adopted to conduct a parametric study to acquire further FE data over a broader spectrum of material strengths and cross-section slendernesses. Based on the combined set of test and FE results, the general design provisions for concrete-filled carbon steel members in the current European and American design codes were evaluated for their applicability to the studied CFDST cross-sections. Overall, the results revealed that both of the examined design codes yield unduly conservative (less so for the higher concrete grades) and scattered moment resistance predictions, though some moment resistances predicted from the European code were on the unsafe side. Modifications, including a concrete reduction factor η to reflect the reduced relative effectiveness of using higher concrete grades and a modified stress distribution considering the partial spread of plasticity, were proposed and shown to improve the consistency of the resistance predictions. Finally, the reliability of the current and modified design rules was demonstrated through statistical analyses.

Flexural strengthening mechanism of concrete-filled steel tube beam by welding round steel at soffit of the beam

In the present study, the flexural behavior of concrete-filled steel tube (CFST) beams strengthened by welding round steel at the soffit of the beam has been investigated. First, a total of six specimens were tested under four-point bending load, which includes one reference beam and five retrofitted beams strengthened by welding round steel of different diameters at the soffit of the beams. Then, the corresponding nonlinear finite-element models have been developed and calibrated against the experimental results. By combining indoor experiments and numerical simulations, the bending moment versus mid-span curvature curves, the flexural capacities, flexural stiffness, ductility, vertical deflection curves, strain distribution across the cross-section, and neutral axis offset of the test beams have been systematically analyzed. The results show that the welded round steel at the soffit can improve the flexural bearing capacity and stiffness of the CFST beams significantly. This can be attributed to the CFST beams benefitting from the flexural bearing effect of the round steel and the downward offset of the neutral axis which causes more core concrete to be in compression. Further, the larger the diameter of the round steel, the greater the benefit. Finally, comparisons have been made with the flexural stiffness calculated using the existing design codes, such as AIJ standard, BS5400, Eurocode 4 and AISC specification. The comparisons show that the flexural stiffness of the unstrengthened CFST beam can be predicted by AIJ and BS5400 perfectly, whereas those of the strengthened CFST beams can only be accurately calculated by AISC.

The Möbius Curvature of Bezier Curves

The aim of this study is to observe the Möbius curvature is computed by me as using curvature of Bezier curve is therefore proportional to the differentials of the curvature also correspond to a such as survey properties of Bezier curves. The Möbius curvature of Bezier curve has different value according to the control points. Also when the different cases may ocur, it has different values according to the angle is constant or not.

Joint estimation of monotone curves via functional principal component analysis

A functional data approach is developed to jointly estimate a collection of monotone curves that are irregularly and possibly sparsely observed with noise. In this approach, the unconstrained relative curvature curves instead of the monotone-constrained functions are directly modeled. Functional principal components are used to describe the major modes of variations of curves and allow borrowing strength across curves for improved estimation. A two-step approach and an integrated approach are considered for model fitting. The simulation study shows that the integrated approach is more efficient than separate curve estimation and the two-step approach. The integrated approach also provides more interpretable principle component functions in an application of estimating weekly wind power curves of a wind turbine.

An efficient corner detector based on ratio of parallelogram diagonal

Corner is an important local feature of image which has been widely applied on various computer vision and image processing tasks. Here, a contour-based corner detector is developed by using the ratio of parallelogram diagonals (RPD) to estimate the curve curvature. The main advantage of RPD detector is that only once square root operation is required to calculate the curvature value at each point on a contour while maintaining good noise robustness. The contributions of this paper include the following three aspects: First, the motivation of the proposed RPD curvature is illustrated by means of parallelogram theory; second, a complete corner detector is proposed based on RPD curvature; third, comprehensive experiments are carried out and the experimental results show superior performances of the proposed method against another five strong baselines. In these experiments, RPD runs 100% faster than the prior works. Moreover, a mean accuracy of 83.87% is reported on GCM dataset which is an improvement of about 0.9% and a mean accuracy of 74.21% is reported on CPDA dataset which is an improvement of about 0.2%.

On admissible curves and their evolution equations in pseudo-Galilean space

The evolution equations of some forms of admissible curves in the pseudo-Galilean Space \(G_{3}^1\) are investigated in this paper. In more detail, we use two separate methods to obtain coupled nonlinear partial differential equations of time evolution in terms of their curvatures. The first method studies the evolution equations for admissible curves via the frame field, while the second studies the evolution equations via the velocity vector. Then, the position vectors of the evolving curves are formulated. Also, we conduct comparative research of the evolution equations for curves in different spaces. We furthermore present some models as an application of the evolution equations of the curvature and torsion for admissible curves, confirming our theoretical results.

The Sub-Riemannian Limit of Curvatures for Curves and Surfaces and a Gauss-Bonnet Theorem in the Group of Rigid Motions of Minkowski Plane with General Left-Invariant Metric

The group of rigid motions of the Minkowski plane with a general left-invariant metric is denoted by E 1 , 1 , g λ 1 , λ 2 , where λ 1 ≥ λ 2 &gt; 0 . It provides a natural 2 -parametric deformation family of the Riemannian homogeneous manifold Sol 3 = E 1 , 1 , g 1 , 1 which is the model space to solve geometry in the eight model geometries of Thurston. In this paper, we compute the sub-Riemannian limits of the Gaussian curvature for a Euclidean C 2 -smooth surface in E 1 , 1 , g L λ 1 , λ 2 away from characteristic points and signed geodesic curvature for the Euclidean C 2 -smooth curves on surfaces. Based on these results, we get a Gauss-Bonnet theorem in the group of rigid motions of the Minkowski plane with a general left-invariant metric.

Algorithm for Determining the Curvature of the Project Line of a Truck Haul Road and the Rate of Change in Its Curvature

It is recognized that a visually smooth and clear road, successfully integrated into the landscape, providing a constant or smoothly variable traffic mode, reduces the tension and fatigue of drivers, thereby contributing to their efficiency. That also reduces transport costs by choosing the most optimal traffic mode. These qualities can be achieved through spatial design method, which consists in creating conditions for safe driving at high speeds. Until now, the spatial road design method has been mainly based on the empirical rules of tracing when using visual images or models for the smoothness and clarity of the designed route of a truck haul road. Based on this method, one makes the necessary corrections to the road by evaluating the original images. Therefore, the effectiveness of the methodological foundations of spatial design of truck haul roads is further increased by providing clarity and visually acceptable curvature of spatial curves, which requires significant development and continued search to find new effective solutions. The work aims to develop an algorithm to evaluate the visual smoothness and clarity of the project line curvature of a truck haul road and the rate of change in its curvature. The article reveals a set of quantitative indicators that sufficiently and completely characterize the visual smoothness and clarity of the central projections of elementary spatial and plane curves. The presented algorithm and the indicators determined based on this algorithm; allow estimating both the visual smoothness and clarity of curves of truck haul roads. The proposed recommendations for the design of spatial curves are checked and refined based on the developed algorithm.

Out of plane performance of novel concrete sandwich panel using different geosynthetics

The main objective of the study is to investigate the flexural behaviour of concrete sandwich panels strengthened with geosynthetics. Two types of geosynthetic material are being used viz. plastic geogrid and polyester geogrid. An attempt has also been made to enhance the strength by using two different mixes. The panels are tested under a monotonically increasing load using a hydraulic jack. The load–deflection curves of the panels were examined to evaluate the effect of geosynthetics on the strength, stiffness, and deformability of panels. The result shows that plastic geogrid reinforced sandwich panels enhanced the flexural strength and ductility of panels compared with control specimens. Polyester geo-grid as reinforcement is also found to be relatively better in restoring the deformability and energy absorption of panels. The results show a gain of about 28% in plastic geogrid (SPGG); 36.8% in the case of polyester geogrid (SGi200), in comparison with the control specimen. The moment–curvature curves were also examined.

Machine learning algorithm improves accuracy of ortho-K lens fitting in vision shaping treatment

PurposeTo construct a machine learning (ML)-based model for estimating the alignment curve (AC) curvature in orthokeratology lens fitting for vision shaping treatment (VST), which can minimize the number of lens trials, improving efficiency while maintaining accuracy, with regards to its improvement over a previous calculation method. MethodsData were retrospectively collected from the clinical case files of 1271 myopic subjects (1271 right eyes). The AC curvatures calculated with a previously published algorithm were used as the target data sets. Four kinds of machine learning algorithms were implemented in the experimental analyses to predict the targeted AC curvatures: robust linear regression models, support vector machine (SVM) regression models with linear kernel functions, bagging decision trees, and Gaussian processes. The previously published calculation method and the novel machine learning method were then compared to assess the final parameters of ordered lenses. ResultsThe linear SVM and Gaussian process machine learning models achieved the best performance. The input variables included sex, age, horizontal visible iris diameter (HVID), spherical refraction (SER), cylindrical refraction, eccentricity value (e value), flat K (K1) and steep K (K2) readings, anterior chamber depth (ACD), and axial length (AL). The R-squared values for the output AC1K1, AC1K2 and AC2K1 values were 0.91, 0.84, and 0.73, respectively. The previous calculation method and machine learning methods displayed excellent consistency, and the proposed methods performed best based on flat K reading and e values. ConclusionsThe ML model can provide practitioners with an efficient method for estimating the AC curvatures of VST lenses and reducing the probability of cross-infection originating from trial lenses, which is especially useful during pandemics, such as that for COVID-19.

Applications of fractional calculus in equiaffine geometry: plane curves with fractional order

In this paper, we introduce the notions of equiaffine arclength and curvature with fractional order for a plane curve and compare them with the standard ones. By using these definitions, we obtain an equiaffine Frenet formula and then construct an analogue of the fundamental theorem. The plane curves of constant equiaffine curvature with fractional order are classified. Several examples are also illustrated.

Construction of a curve by using the state equation of Frenet formula

Abstract In this paper, the available formulae for the curvature of plane curve are reviewed not only for the time-like but also for the space-like parameter curve. Two ways to describe the curve are proposed. One is the straight way to obtain the Frenet formula according to the given curve of parameter form. The other is that we can construct the curve by solving the state equation of Frenet formula subject to the initial position, the initial tangent, normal and binormal vectors, and the given radius of curvature and torsion constant. The remainder theorem of the matrix and the Cayley–Hamilton theorem are both employed to solve the Frenet equation. We review the available formulae of the radius of curvature and examine their equivalence. Through the Frenet formula, the relation among different expressions for the radius of curvature formulae can be linked. Therefore, we can integrate the formulae in the engineering mathematics, calculus, mechanics of materials and dynamics. Besides, biproduct of two new and simpler formulae and the available four formulae in the textbook of the radius of curvature yield the same radius of curvature for the plane curve. Linkage of centrifugal force and radius of curvature is also addressed. A demonstrative example of the cycloid is given. Finally, we use the two new formulae to obtain the radius of curvature for four curves, namely a circle. The equivalence is also proved. Animation for 2D and 3D curves is also provided by using the Mathematica software to demonstrate the validity of the present approach.

Clothoid fitting and geometric Hermite subdivision

We consider geometric Hermite subdivision for planar curves, i.e., iteratively refining an input polygon with additional tangent or normal vector information sitting in the vertices. The building block for the (nonlinear) subdivision schemes we propose is based on clothoidal averaging, i.e., averaging w.r.t. locally interpolating clothoids, which are curves of linear curvature. To this end, we derive a new strategy to approximate Hermite interpolating clothoids. We employ the proposed approach to define the geometric Hermite analogues of the well-known Lane-Riesenfeld and four-point schemes. We present numerical results produced by the proposed schemes and discuss their features.

An Improved Prediction Method of Soil-Water Characteristic Curve by Geometrical Derivation and Empirical Equation

Much attention has been paid on the soil-water characteristic curve (SWCC) during decades because it plays great roles in unsaturated soil mechanics. However, it is time-consuming and costly to obtain a series of entire saturation-suction data by experiments. The curves acquired by directly fitting empirical equations to limited experimental data are greatly different from the actual SWCC, and the relevant soil parameters obtained by inaccurate curve are also incorrect. Thus, an improved prediction method for more accurate entire SWCC was established. This novel method was based on the analysis of shape characteristics of SWCC with three critical points S , C 1 , and C 2 under the hypothesis of geometrical symmetric relation. The theoretical computation was specifically deduced under conventional Gardner, VG, and FX models, respectively, and then inferred on different soil types of 45 collected SWCC datasets. This geometrical symmetric relation exhibited well in all these three conventional empirical equations, especially in Gardner equation. Finally, a series of filer paper tests on sand, silt, and clay were also carried out to acquire entire SWCC curve for the verification and evaluation of the proposed geometrical method. Results show that this improved prediction method effectively decreases deviation resulting from directly fitting empirical equations to limited data of wide types of soils. The averaged improvement was larger under VG equation than under Gardner and FX equation. It proved that the accuracy of predicting greatly depends on the shape characteristic point of maximum curve curvature (point C 2 ), other than the number of points. This research provides a novel computation method to improve prediction accuracy even under relative less experimental data.

The Sub-Riemannian Limit of Curvatures for Curves and Surfaces and a Gauss–Bonnet Theorem in the Rototranslation Group

The rototranslation group ℛ T is the group comprising rotations and translations of the Euclidean plane which is a 3-dimensional Lie group. In this paper, we use the Riemannian approximation scheme to compute sub-Riemannian limits of the Gaussian curvature for a Euclidean C 2 -smooth surface in the rototranslation group away from characteristic points and signed geodesic curvature for Euclidean C 2 -smooth curves on surfaces. Based on these results, we obtain a Gauss–Bonnet theorem in the rototranslation group.

Correction to: The total intrinsic curvature of curves in Riemannian surfaces

Correction to: The total intrinsic curvature of curves in Riemannian surfaces

Parallel curves

Properties of curved functions considered to be parallel over their domains are investigated. Parallel curves in a given plane may appear identical but are actually not superimposable and thus are not congruent. Translational shifted functions in a plane are not parallel curves because the shortest perpendicular distance between them is not constant along the curves. It is generally not possible to superimpose parallel curves by simple translational shifting in a given 2-dimensional plane. Although the first derivative of parallel curves in a given plane produces equal magnitudes at points where perpendicular line segments or vectors cross the curves, the second derivatives must always differ in magnitude at these positions. The curvature of parallel curves differs, while the slopes of the curves are identical at corresponding points on the curves where perpendicular segments cross. Examples of parallel curves in nature are presented, and curve fitting is used to estimate the formula for a curve that is parallel to a parabola in the same plane.