Arc Length Parameter Research Articles

BMO embeddings, chord-arc curves, and Riemann mapping parametrization

We consider the space of chord-arc curves on the plane passing through infinity with their parametrization γ defined on the real line, and embed this space into the product of the BMO Teichmüller spaces. The fundamental theorem we prove about this representation is that γ↦log⁡γ′ is a biholomorphic homeomorphism into the complex Banach space of BMO functions. Using these two equivalent complex structures, we develop a clear exposition on the analytic dependence of involved mappings between certain subspaces. Especially, we examine the parametrization of a chord-arc curve by using the Riemann mapping and its dependence on the arc-length parametrization. As a consequence, we can solve the conjecture of Katznelson, Nag, and Sullivan by showing that this dependence is not continuous.

Notes on Curves at a Constant Distance from the Edge of Regression on a Curve in Galilean 3-Space

In this paper, we define the curve at a constant distance from the edge of regression on a curve r(s) with arc length parameter s in Galilean 3-space. Here, d is a non-isotropic or isotropic vector defined as a vector tightly fastened to Frenet trihedron of the curve r(s) in 3-dimensional Galilean space. We build the Frenet frame of the constructed curve with respect to two types of the vector d and we indicate the properties related to the curvatures of the curve . Also, for the curve , we give the conditions to be a circular helix. Furthermore, we discuss ruled surfaces of type A generated via the curve and the vector D which is defined as tangent of the curve in 3-dimensional Galilean space. The constructed ruled surfaces also appear in two ways. The first is constructed with the curve and the non-isotropic vector D. The second is formed by the curve and the non-isotropic vector D. We calculate the distribution parameters of the constructed ruled surfaces and we show that the ruled surfaces are developable. Finally, we provide examples and visuals to back up our research.

Equidistant Tool Path and Cartesian Trajectory Planning for Robotic Machining of Curved Freeform Surfaces

This article proposes an equidistant tool path planning strategy on curved freeform surfaces with the focus on robotic machining tasks. The strategy is based on the arc-length parameterization of the surface and can be considered as a variant of the iso-parametric path planning. This method allows us to directly consider a given Cartesian boundary in the parameter domain. Using an inverse interpolation scheme of the parameterization with cubic splines, the path planned in arc-length coordinates is mapped back in Cartesian coordinates. The equidistance property is also satisfied for larger path intervals and does not depend on the local curvature of the surface. The proposed trajectory planner ensures a constant velocity along the planned path on the freeform surface while limiting the maximum centripetal acceleration. However, general velocity profiles can be included depending on the actual machining application. Exploiting the differential geometric properties of the path and surface allows us to directly determine the orientation and angular velocity of the end-effector along the planned trajectory. The proposed methods are demonstrated with a robotic manipulator on a real curved freeform surface. Note to Practitioners—Robotic manipulators and five-axis computerized numerical control (CNC) machines provide high flexibility for machining processes on curved freeform surfaces, for example, polishing or grinding. For these applications, it is often desired that the distance between two subsequent machining paths is independent of the local geometry and surface curvature. This article proposes a novel method to plan equidistant coverage paths on a curved freeform surface. Therefore, we develop a transformation between the 3-D Cartesian coordinates and the 2-D parameters based on the arc-lengths along the surface. The geometry is given by CAD data or a 3-D scan of the part and is directly processed, making our approach particularly feasible for both large to small batch sizes and even individualized products. To cover an arbitrary area with parallel paths, its boundary—given in Cartesian coordinates—is transformed into arc-length parameters. The coverage path is planned in this parameter space, consisting of parallel, equidistant lines with interconnection paths. The path is transformed back in Cartesian coordinates using an inverse transformation. We plan the motion of the tool along this path, satisfying a desired feed rate depending on the actual machining application. The tool is slowed down when the maximum centripetal acceleration is reached. Considering the geometry of the path and the surface, we can also plan the tool’s orientation and its angular velocity during the manufacturing task. We demonstrate the path and motion planning approach on an experimental setup with a robotic manipulator.

A C2 continuous trajectory planning method for 6-DOF rotational robot manipulators

PurposeThe purpose of this paper is to propose a smooth double-spline interpolation method for six-degree-of-freedom rotational robot manipulators, achieving the global C2 continuity of the robot trajectory.Design/methodology/approachThis paper presents a smooth double-spline interpolation method, achieving the global C2 continuity of the robot trajectory. The tool center positions and quaternion orientations are first fitted by a cubic B-spline curve and a quartic-polynomial-based quaternion spline curve, respectively. Then, a parameter synchronization model is proposed to realize the synchronous and smooth movement of the robot along the double spline curves. Finally, an extra u-s function is used to record the relationship between the B-spline parameter and its arc length parameter, which may reduce the feed rate fluctuation in interpolation. The seven segments jerk-limited feed rate profile is used to generate motion commands for algorithm validation.FindingsThe simulation and experimental results demonstrate that the proposed method is effective and can generate the global C2-continuity robot trajectory.Originality/valueThe main contributions of this paper are as follows: guarantee the C2 continuity of the position path and quaternion orientation path simultaneously; provide a parameter synchronization model to realize the synchronous and smooth movement of the robot along the double spline curves; and add an extra u-s function to realize arc length parameterization of the B-spline path, which may reduce the feed rate fluctuation in interpolation.

Optical modeling for recursional Hasimoto map with normalized time fractional applications

In this paper, we construct the optical flow equations obtained with the help of a timelike curve in De Sitter 2D space S12 and with the aid of the Hasimoto map defined by this curve. First, we introduce the Hasimoto map, which we call the De-Hasimoto map, for a timelike curve in S12. Then, with the help of this map and the equation representing the vortex filament motion, we obtain some flow equations with time and arc length parameters. Finally, we compute exact solutions of time fraction of an obtained optical motion equation, and we present some physical reviews on it.

Building Information Protection Method of Urban Historical Features Based on BIM Technology

In order to improve the building information protection method of urban historical features, this paper combines BIM technology to study the building information protection method of urban historical features and studies the least-squares fitting technology of NURBS curves. On this basis, this paper proposes a fitting method of the constrained least-squares method. Moreover, this paper designs a constrained least-squares method for the requirement of fitting accuracy and gives a method for setting the initial value. In addition, this paper proposes a NURBS interpolation method based on arc length parameter compensation technology. The experimental study verifies that the building information protection method of urban historical features based on BIM technology has a good effect and can effectively promote the building information protection of urban historical features.

Numerical Reparametrization of Periodic Planar Curves Via Curvature Interpolation

A novel static algorithm is proposed for numerical reparametrization of periodic planar curves. The method identifies a monitor function of the arclength variable with the true curvature of an open planar curve and considers a simple interpolation between the object and the unit circle at the curvature level. Since a convenient formula is known for tangential velocity that maintains the equidistribution rule with curvature-type monitor functions, the strategy enables to compute the correspondence between the arclength and another spatial variable by evolving the interpolated curve. With a certain normalization, velocity information in the motion is obtained with spectral accuracy while the resulting parametrization remains unchanged. Then, the algorithm extracts a refined representation of the input curve by sampling its arclength parametrization whose Fourier coefficients are directly accessed through a simple change of variables. As a validation, improvements to spatial resolution are evaluated by approximating the invariant coefficients from downsampled data and observing faster global convergence to the original shape.

3D curve regularization

In this paper, we study the regularization of 3D curves connecting two points. We propose an energy-based formulation which is an extension to 3D of the geodesic active contours introduced in 2D by Caselles et al. in 1997. By minimizing this energy we try to minimize the curve length but keeping the curve close to the original one. The energy depends on a regularization parameter which determines the smoothness of the regularized curve. We show the Euler-Lagrange equation of the proposed energy using the arc-length parameterization of the curve. We interpret the Euler-Lagrange equation in terms of the Frenet–Serret frame and we study some qualitative properties of the energy minima. We apply the steepest-descent method to approximate the local minima of the energy using an evolution equation. We propose a numerical scheme to solve the evolution equation and we present some experiments on real data in the context of aortic centerline regularization.

Spinor Equations of Successor Curves

The aim of this study is to give spinor representation of successive curves in three-dimensional Euclidean space. In three dimensional Euclidean Space, the spinor representations of a curve with unit speed and a successor curve with the same arc length parameter as this curve has been studied. For this, first of all, the curve with unit speed and its successor curve have been corresponded to two different spinors. Then, using the relationships between these curves, the relationships between the spinors corresponding to these curves have been given. Therefore, geometric interpretations of these curves and corresponding spinors have been made. In addition, different spinor equations of the mates and derivatives of spinors have been examined and geometric interpretations of these spinor equations have been given. Then, spinor equations have been obtained in case the successive curves are helices. Consequently, two examples have been given.

The Two-Dimensional Liquid Crystal Droplet Problem with a Tangential Boundary Condition

This paper studies a shape optimization problem which reduces to a nonlocal free boundary problem involving perimeter. It is motivated by a study of liquid crystal droplets with a tangential anchoring boundary condition and a volume constraint. We establish in 2D the existence of an optimal shape that has two cusps on the boundary. We also prove that the boundary of the droplet is a chord–arc curve with its normal vector field in the VMO space, and its arc-length parameterization belongs to the Sobolev space $$H^{3/2}$$ . In fact, the boundary curves of such droplets closely resemble the so-called Weil–Petersson class of planar curves. In addition, the asymptotic behavior of the optimal shape when the volume becomes extremely large or small is studied.

A Novel FFHE-Inspired Method for Large Power System Static Stability Computation

This article proposes a novel method termed HEAP for fast and reliably computing nose points, tracing the entire PV curve, and assessing static stability limits. Here, the prefix “HE” in HEAP refers to (fast and flexible) <underline xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">h</u> olomorphic <underline xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e</u> mbedding, and “AP” is short for <underline xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a</u> rc-length <underline xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</u> arametrization. This novel method uses an arc-length parametrization and piecewise approximants that enable it to have the favorable feature of Continuation Power Flow that can pass through the nose point without numerical difficulties. Another outstanding feature of the proposed method HEAP is its computational efficiency by enabling the accurate approximation over a long interval of the parameter (i.e., a large step size). This is because of the large convergence region of HEAP. The existing methods for similar tasks usually suffer from several issues such as empirical criteria for detecting nose points, small convergence regions of correctors, numerous intermediate points, and large memory space consumption. These issues are well addressed by HEAP. To examine its numerical performance, HEAP is tested on various power system models with as many as 70,000 buses, and numerical results substantiate the outstanding features of HEAP. Depending on intended applications (e.g., only locating the nose point or fully tracking the curve), two tailored approaches are devised to effectively derive numerical solutions. In comparison with the continuation-based power flow method, HEAP attains speedups of 3x to 300x and notably attains speedups of at least 19x for all large test cases with 10,000 or more buses. The continuation-based power flow method fails in several large test cases, however, all of which are successfully solved by HEAP. After enforcing the generator reactive power limit, HEAP also produces the same accurate results as the continuation-based power flow method.

Some Characterizations of Spherical Indicatrix Curves Generated by Flc Frame

In this study, the spherical indicatrices of Flc frame vectors were defined on unit sphere. The arc length parameters and the Frenet vectors of these indicatrix curves were calculated, as well. Last, we have provided the geodesic curvatures according to both Euclidean space $E^3$ and unit sphere $S^2$.

An energy stable finite difference method for anisotropic surface diffusion on closed curves

In this paper, we develop an energy stable finite difference method for the problem of curve motion under anisotropic surface diffusion. The motion of curve by anisotropic surface diffusion is governed by the fourth-order (highly nonlinear) geometric evolution equation. As in Li and Bao (2021), we first split the fourth-order evolution equation into two second-order equations, where the position vector of curve and the weighted curvature are treated as unknowns. Instead of using the arclength parameter, we introduce a Lagrangian coordinate parameter such that a closed curve can be parametrized over a fixed interval so that the equations can be represented using the new parameter. We then propose a linearly semi-implicit finite difference method to discretize these two equations, and prove that the scheme satisfies discrete energy dissipation so it is energy stable under suitable condition on the anisotropic surface energy. To show the applicability of our present method, we perform several numerical tests on various initial curves and different anisotropic energies. The numerical results show that our scheme is indeed energy dissipative and conserves even the total area well.

Characterization of Rectifying Curves by Their Involutes and Evolutes

A rectifying curve is a twisted curve with the property that all of its rectifying planes pass through a fixed point. If this point is the origin of the Cartesian coordinate system, then the position vector of the rectifying curve always lies in the rectifying plane. A remarkable property of these curves is that the ratio between torsion and curvature is a nonconstant linear function of the arc-length parameter. In this paper, we give a new characterization of rectifying curves, namely, we prove that a curve is a rectifying curve if and only if it has a spherical involute. Consequently, rectifying curves can be constructed as evolutes of spherical twisted curves; we present an illustrative example of a rectifying curve obtained as the evolute of a spherical helix. We also express the curvature and the torsion of a rectifying spherical curve and give necessary and sufficient conditions for a curve and its involute to be both rectifying curves.

Effect of dark energy models on the energy content of charged and rotating black holes

The energy content of the charged-Kerr (CK) spacetime surrounded by dark energy (DE) is investigated using approximate Lie symmetry methods for the differential equations. For this, we consider three different DE scenarios: cosmological constant with an equation of state parameter , quintessence DE with an equation of state parameter , and a frustrated network of cosmic strings with an equation of state parameter . To study the gravitational energy of the CK black hole surrounded by the DE, we explore the symmetries of the 2nd-order perturbed geodesic equations. It is noticed, for all the values of ω, the exact symmetries are recovered as 2nd-order approximate trivial symmetries. These trivial approximate symmetries give the rescaling of arc length parameter s in this spacetime which indicates that the energy in the underlying spacetime has to be rescaled by a factor that depends on the black hole parameters and the DE parameter. This rescaling factor is compared with the factor of the CK spacetime found in [Hussain et al. Gen. Relativ. Gravit. (2009)] and the effects of the DE on it are discussed. It is observed that for all the three values of the equation of state parameter ω, the effect of DE results in decreased energy content of the black hole spacetime, regardless of values of the charge Q, spin a and the DE parameter α. This reduction in the energy content due to the involvement of the DE favours the idea of mass reduction of black holes by accretion of DE given by [Babichev et al. Phys. Rev. Lett. (2004)].

High order pseudo arc-length method for strong discontinuity of detonation wave

In this paper, in order to improve the resolution of capturing discontinuities, we introduce the pseudo arc-length parameter to make the mesh move to the discontinuities adaptively. By combining the high-precision WENO scheme with the pseudo arc-length algorithm, the advantages of both schemes can be shown, on the one hand, the solution has a higher convergence rate, on the other hand, it has a higher resolution for the region solution with larger physical variation . Because the traditional high-order scheme is based on Cartesian grid, and the grid in the pseudo arc-length numerical calculation is deformed. In view of the non-uniform grid and non-orthogonal deformation grid caused by the grid moving, the original deformed physical space is mapped to the uniform orthogonal arc-length calculation space by introducing coordinate transformation, and then the classical higher order scheme is used to solve the governing equations in the computational coordinate system. Through the comparison of some numerical examples and the analysis of numerical errors, it can be found that the pseudo arc-length algorithm is better than the finite volume method with fixed mesh. The high-order pseudo arc-length algorithm has a very high resolution to capture discontinuities, and the density of the grid near the discontinuities is very high. The adaptive grid movement weakens the singularity of the governing equation near the discontinuity, so the whole solution is smooth and the numerical oscillation is not obvious. This shows that the pseudo arc-length algorithm can overcome the shortcomings of high-order schemes which easily cause numerical oscillations. Finally, the chemical reaction flow problem is calculated. The results show that the high-order pseudo arc-length numerical algorithm with less mesh number has faster convergence rate and higher discontinuous resolution. Therefore, the high-order pseudo arc-length algorithm has obvious advantages in dealing with the strong discontinuity problem of explosion and shock.

A Cooks Distance Approach to Control the Haz and Arc Length of Tig Mild Steel Weld

The heat affected zone and arc length parameters have a vital role to play in determining the integrity of a weld structure. The cooks distance is a statistical diagnostic employed in this study to select the best optimum combination of welding process parameters. Mild steel plate was the choice material used to produce the weld specimen, which was welded with the Tungsten inert gas method. The RSM model was used to develop an optimal solution that can explain the behavior of the welded joint with respect to the heat affected zone and arc length, different diagnostic techniques were employed which includes the normal probability plot and cooks distance plot. The model developed has sufficient merit as the results obtained shows that the cooks distance values is within the range of 0 and 1 indicating the absence of outlier in the data making the optimal solution highly acceptable.

A Cooks Distance Approach to Control the Haz and Arc Length of Tig Mild Steel Weld

The heat affected zone and arc length parameters have a vital role to play in determining the integrity of a weld structure. The cooks distance is a statistical diagnostic employed in this study to select the best optimum combination of welding process parameters. Mild steel plate was the choice material used to produce the weld specimen, which was welded with the Tungsten inert gas method. The RSM model was used to develop an optimal solution that can explain the behavior of the welded joint with respect to the heat affected zone and arc length, different diagnostic techniques were employed which includes the normal probability plot and cooks distance plot. The model developed has sufficient merit as the results obtained shows that the cooks distance values is within the range of 0 and 1 indicating the absence of outlier in the data making the optimal solution highly acceptable.

A New Class of Plane Curves with Arc Length Parametrization and Its Application to Linear Analysis of Curved Beams

The objective of this paper is to define one class of plane curves with arc-length parametrization. To accomplish this, we constructed a novel class of special polynomials and special functions. These functions form a basis of L2(R) space and some of their interesting properties are discussed. The developed curves are used for the linear static analysis of curved Bernoulli–Euler beam. Due to the parametrization with arc length, the exact analytical solution can be obtained. These closed-form solutions serve as the benchmark results for the development of numerical procedures. One such example is provided in this paper.

StrokeStrip

When creating freeform drawings, artists routinely employ clusters of overdrawn strokes to convey intended, aggregate curves. The ability to algorithmically fit these intended curves to their corresponding clusters is central to many applications that use artist drawings as inputs. However, while human observers effortlessly envision the intended curves given stroke clusters as input, existing fitting algorithms lack robustness and frequently fail when presented with input stroke clusters with non-trivial geometry or topology. We present StrokeStrip , a new and robust method for fitting intended curves to vector-format stroke clusters. Our method generates fitting outputs consistent with viewer expectations across a vast range of input stroke cluster configurations. We observe that viewers perceive stroke clusters as continuous, varying-width strips whose paths are described by the intended curves. An arc length parameterization of these strips defines a natural mapping from a strip to its path. We recast the curve fitting problem as one of parameterizing the cluster strokes using a joint 1D parameterization that is the restriction of the natural arc length parameterization of this strip to the strokes in the cluster. We simultaneously compute the joint cluster parameterization and implicitly reconstruct the a priori unknown strip geometry by solving a variational problem using a discrete-continuous optimization framework. We use this parameterization to compute parametric aggregate curves whose shape reflects the geometric properties of the cluster strokes at the corresponding isovalues. We demonstrate StrokeStrip outputs to be significantly better aligned with observer preferences compared to those of prior art; in a perceptual study, viewers preferred our fitting outputs by a factor of 12:1 compared to alternatives. We further validate our algorithmic choices via a range of ablation studies; extend our framework to raster data; and illustrate applications that benefit from the parameterizations produced.