Space Curves Research Articles

Pure shape dynamics: general framework

We put forward a general framework for describing relational physical theories, which we call pure shape dynamics (PSD). Elaborating on the original insights brought about by the shape dynamics programme, PSD’s novel take on relationalism is its insistence on describing any dynamical system by means of the intrinsic geometry of its associated curve in the suitable relational configuration space of the theory, namely shape space, whereby the corresponding equation of state of the curve expresses the ratio of change of one of its geometric degrees of freedom with respect to another one. The mathematical structure underlying the equation of state is a local section over a natural generalisation of the unit tangent bundle, which we call shape phase space.

Deployment of second-order networked mobile agents over a smooth curve

In this paper, we investigate second-order deployment of multiple mobile agents over a C2 curve in 2D or 3D space. We assume that an informed agent in the group can measure its relative position with respect to the given curve and velocity, other agents have access to the local information about the given curve as well as the position and velocity in regard to their closest neighbors, where the agent dynamics is described by a strongly damped wave equation. Without delay, we establish a boundary control law for the informed agent. When taking into account time-varying delay, we assume that the informed agent transmits its measurements to its neighbors through a communication mechanism. Given appropriate controller gains, second-order deployment can be achieved with an exponential decay. Finally, some simulation results are presented to illustrate our theoretical results.

Hybrid Full Adders: Optimized Design, Critical Review and Comparison in the Energy-Delay Space

In this paper, we design and compare seven meaningful hybrid one-bit full adders topologies that are optimized in terms of energy-delay trade-offs to operate in multibit ripple carry adders. The goal is to provide the designer with a simple and powerful approach for choosing the best topology for a given power budget, speed performance, or any combination of both. The design and comparison deal with 4-bit and 8-bit ripple carry adders and exploit the derivation of the energy-efficient curves in the energy-delay space. To do so, first we define the procedures to obtain energy consumption and propagation delay by simulating a ripple carry adder designed at a transistor level. Then, we introduce a design methodology to optimize a ripple carry adder by minimizing some significant figures-of-merit in terms of energy-delay trade-offs. The comparison of the energy-efficient curves allows us to make a simple and effective comparison as well as to identify the best one-bit full adder topologies.

The Phase Space Analysis of Interacting K-Essence Dark Energy Models in Loop Quantum Cosmology

The present work deals with two kinds of k-essence dark energy models within the framework of loop quantum cosmology (LQC). The two kinds of k-essence models originates from two forms of Lagrangians, i.e., L1=F(X)V(ϕ) and L2=F(X)−V(ϕ), where F(X) and V(ϕ) stand for the kinetic term and potential of the scalar field ϕ, respectively. Two models are based on different phase variables settings, and the general form of autonomous dynamical system is deduced for each Lagrangian. Then, the dynamical stabilities of the critical points in each model are analysed in different forms of F(X) and V(ϕ). Model I is a 3-dim system with four stable points, and Model II is a 4-dim system but reduced to a 3-dim system using the symmetry analysis, which has five stable points. Moreover, the corresponding cosmological quantities, such as Ωϕ, wϕ and q, are calculated at each critical point. To compare these with the case of the classical Einstein cosmology (EC), the dynamical evolutionary trajectories in the phase space and evolutionary curves of the cosmological quantities are drawn for both EC and LQC cases, which shows that the loop quantum gravity effects diminish in the late-time universe but are significant in the early time. Further, the effects of interaction Q=αHρm on the evolutions of the universe are discussed. With the loop quantum gravity effects, bouncing universe is achieved in both models for different initial values of ϕ0, ϕ˙0, H0, ρ0 and coupling parameter α, which helps to avoid singularities. However, the interaction has little effect on bounce, although it is important to the stability of some critical points.

3-Boyutlu Galilean Uzayında Tüp Yüzeyler Üzerine Bir Araştırma

In this study, the tube surfaces generated by the curve defined in Galilean 3-space are examined and some certain results of describing the geodesics on the surfaces are also given. Furthermore, the conditions of being geodesic on the tubular surface are obtained with the help of Clairaut’s theorem, which allows us to constitute the specific energy. The physical meaning of the specific energy and the angular momentum is of course related with the physical meaning itself. Our results show that the specific energy and the angular momentum obtained on tubular surfaces can be expressed using arbitrary geodesic curve in Galilean space. In addition, some characterizations are given for these surfaces, with the obtained mean and Gaussian curvatures.

Committor-Consistent Variational String Method.

The treatment of slow and rare transitions in the simulation of complex systems poses a great computational challenge. A powerful approach to tackle this challenge is the string method, which represents the transition path as a one-dimensional curve in a multidimensional space of collective variables. Commonly used strategies for pathway optimization include aligning the tangent of the string to the local mean force or to the mean drift determined from swarms of short trajectories. Here, a novel strategy is proposed, allowing the string to be optimized based on a variational principle involving the unidirectional reactive flux expressed in terms of the time-correlation function of the committor. The method is illustrated with model systems and then probed with the alanine dipeptide and a coarse-grained model of the barstar-barnase protein complex. Successive iterations variationally refine the string toward an optimal transition pathway following the gradient of the committor between two metastable states.

A hot sub-Neptune in the desert and a temperate super-Earth around faint M dwarfs

Aims. We report the discovery and validation of two TESS exoplanets orbiting faint M dwarfs: TOI-4479b and TOI-2081b. Methods. We jointly analyzed space (TESS mission) and ground-based (MuSCAT2, MuSCAT3 and SINISTRO instruments) light curves using our multicolor photometry transit analysis pipeline. This allowed us to compute contamination limits for both candidates and validate them as planet-sized companions. Results. We found TOI-4479b to be a sub-Neptune-sized planet (Rp = 2.82−0.63+0.65 R⊕) and TOI-2081b to be a super-Earth-sized planet (Rp = 2.04−0.54+0.50 R⊕). Furthermore, we obtained that TOI-4479b, with a short orbital period of 1.15890−0.00001+0.00002 days, lies within the Neptune desert and is in fact the largest nearly ultra-short period planet around an M dwarf known to date. Conclusions. These results make TOI-4479b rare among the currently known exoplanet population of M dwarf stars and an especially interesting target for spectroscopic follow-up and future studies of planet formation and evolution.

A new approach to the connections between moving curves and a family of complex KdV type systems

In this paper, we give a new approach to the existence of a connection between the geometry of moving curves and the soliton equation for a family of complex KdV type systems by using two other classes of curve evolution in Euclidean space. As an application, we derive the soliton surfaces associated with the single soliton solutions of the given time evolution equations of the curvature function for the two‐dimensional motion of the space curve.

Generalized Bertrand and Mannheim Curves in 3D Lie Groups

In this paper, we give a new approach for Bertrand and Mannheim curves in $3D$ Lie groups with bi-invariant metrics. In this way, some conditions including the known results have been given for a curve to be Bertrand or Mannheim curve in $3D$ Euclidean space and in $3D$ Lie groups.

Optimization of Light Field for Generation of Vortex Knot

The theory of knots and links focuses on the embedding mode of one or several closed curves in three-dimensional Euclidean space. In an electromagnetic field system, all-optical knots or links composed of phase or polarization singularities have been verified theoretically and experimentally. Recent studies have shown that robust topological all-optical coding can be achieved by using optical knots and links. However, in the current design of optical knots and links based on phase or polarization singularities, the amplitude of light between adjacent singularities is relatively weak. This brings great pressure to detection of optical knots and links and limits their applications. Here, we propose a new optimization method in theory. Compared with the existing design methods, our design method improves the relative intensity distribution of light between adjacent singularities. We verify the feasibility of our design results in experiments. Our study reduces the detection difficulty of optical knots and links, and has a positive significance for promotion of applications of optical knots and links.

Evolutes of Fronts in de Sitter and Hyperbolic Spheres

The evolute of a regular curve is a classical object from the viewpoint of differential geometry. We study some types of curves such as framed curves, framed immersion curves, frontal curves and front curves in 2-dimensional de Sitter and hyperbolic spaces. Also, we investigate the evolutes and some of their properties of fronts at singular points under some conditions. Finally, some computational examples in support of our main results are given and plotted.

A unified method for constructing developable surface pencils interpolating characteristic curves

Developable surface is a simple and common surface in surface modeling. Geodesic, line of curvature, asymptotic curve, and D-type curve are important characteristic curves on the surfaces. This study gives a unified method for constructing developable surface pencils interpolating these four kinds of characteristic curves. Given a regular space curve R(r), we derive a new condition that a surface pencil P (r, t) interpolating R(r) is developable. The result shows that the condition completely depends on a univalent function λ and an angle θ. By choosing different λ and θ, we can not only control the shape of P (r, t), but also make R(r) become any kind of characteristic curve on P (r, t). Furthermore, we take natural and conjugate curve pairs as those characteristic curves to construct developable surface pairs. Finally, an example of a slant helix shows that the proposed unified method is more general than other methods, and has good interactivity and convenience.

On Tzitzeica Curves According to Q-Frame in Euclidean 3-Space

Tzitzeica curve is a special spatial curve, which introduced Gheorghe Tzitzeica in 1911. Gheorghe Tzitzeica defined this curve as follows; for a spatial curve ϕ, "if the ratio of its torsion τ and the square of the distance d from the origin to the osculating plane at an arbitrary point of the curve is constant, then this spatial curve is a Tzitzeica curve in Euclidean space." Moreover Gheorghe Tzitzeica defined a special surface which is named Tzitzeica surface in 1907. In this surface, the asymptotic lines of Tzitzeica surfaces with negative Gaussian curvature are Tzitzeica curves. Also, the ratio of its Gaussian curvature K and the distance d from the origin to the tangent plane at any arbitrary point of the surface is constant. In this paper, we study the Tzitzeica curve via q-frame in E^3. Firstly, we redefine the Tzitzeica curve q-frame in Euclidean 3- space. Then, it is obtained some conditions for the Tzitzeica curve as a spherical curve. Finally, we investigate harmonic vector cases of the Tzitzeica curve according to q- frame in Euclidean space.

A Sufficient Condition for a Curve to Belong to Boundary of its Own Convex Hullin An

The paper discusses some generalizations of the convex curve and the relationship between the classes of curves they define.
 The main result concerns the curves in n-dimensional affine space An: a non-degenerate curve in An lies on the boundary of its convex hull if any hyperplane intersects it in at most n points. It is justified by the theory of convex sets. This statement generalizes an earlier (2014) result of the author concerning the curves that are compact sets in An. V. Gustin proved in 1947 a stronger theorem for a Euclidean space En of dimension n: a connected set intersected by any hyperplane at no more than n points is a simple continuous curve lying on the boundary of a convex set. Note that in a projective space of dimension n the requirement that the curve intersects all planes at no more than n points is included in the definition of a convex curve. Thus, the established fact, together with the result of V. Gustin, shows the contiguity of the concepts of convexity in all three spaces.

Zindler-type hypersurfaces in [formula omitted

In this paper the definition of Zindler-type hypersurfaces is introduced in R4 as a generalization of planar Zindler curves. After recalling some properties of planar Zindler curves, it is shown that Zindler hypersurfaces satisfy similar properties. Techniques from quaternions and symplectic geometry are used. Moreover, each Zindler hypersurface is fibrated by space Zindler curves that correspond, in the convex case, to some space curves of constant width lying on the associated hypersurface of constant width and with the same symplectic area.

Cross-section measures, radii, and Radon curves

The purpose of this paper is to study cross-section measures and their connections to Radon curves in two-dimensional Minkowski spaces. More precisely, we use results on cross-section measures to present, for nomed planes, inner and outer radii of the unit ball with respect to its isoperimetrix. This yields new characterizations of and further results on Radon curves.

Path planning of hyper‐redundant manipulators for narrow spaces

Compared with the traditional manipulator, the hyper-redundant manipulator has the advantage of high flexibility, which is particularly suitable for all kinds of complex working environments. However, the complex space environment requires the hyper-redundant manipulator to have stronger obstacle avoidance ability and adaptability. In order to solve the problems of a large amount of calculation and poor obstacle avoidance effects in the path planning of the hyper-redundant manipulator, this paper introduces the ‘backbone curve’ approach, which transforms the problem of solving joint path points into the behaviour of determining the backbone curve. After the backbone curve approach is used to design the curve that meets the requirements of obstacle avoidance and the end pose, the least squares fitting and the improved space joint fitting are used to match the plane curve and the space curve respectively, and the angle value of each joint of the manipulator is limited by the algorithm. Furthermore, a fusion obstacle avoidance algorithm is proposed to obtain the joint path points of the hyper-redundant manipulator. Compared with the classic Jacobian iteration method, this method can avoid obstacles better, has the advantages of simple calculation, high efficiency, and can fully reflect the geometric characteristics of the manipulator. Simulation experiments have proven the feasibility of the algorithm.

Filament plots for data visualization

The efficiency of modern computer graphics allows us to explore collections of space curves simultaneously with “drag-to-rotate” interfaces. This inspires us to replace “scatterplots of points” with “scatterplots of curves” to simultaneously visualize relationships across an entire dataset. Since spaces of curves are infinite dimensional, scatterplots of curves avoid the “lossy” nature of scatterplots of points. In particular, if two points are close in a scatterplot of points derived from high-dimensional data, it does not generally follow that the two associated data points are close in the data space. Andrews plots provide scatterplots of curves that perfectly preserve Euclidean distances, but simultaneous visualization of these graphs over an entire dataset produces “visual clutter” because graphs of functions generally overlap in 2D. We mitigate this “visual clutter” issue by constructing computationally inexpensive 3D extensions of Andrews plots. First, we construct optimally smooth 3D Andrews plots by considering linear isometries from Euclidean data spaces to spaces of planar parametric curves. We rigorously parametrize the linear isometries that produce (on average) optimally smooth curves over a given dataset. This parameterization of optimal isometries reveals many degrees of freedom, and (using recent results on generalized Gauss sums) we identify a particular member of this set which admits an asymptotic “tour” property that avoids certain local degeneracies as well. Finally, we construct unit-length 3D curves (filaments) from Bishop frames induced by 3D Andrews plots. We conclude with examples of filament plots for several standard datasets, 1 illustrating how filament plots avoid “visual clutter”.

Geometric Modeling of Serrated Cutters for Endmilling Forces using Local Oblique Cutting and Rake Normal Correspondence

For nonstandard endmills with sinusoidal and circular serrations on nominally helical cutting edges, the edge space curve was subdivided into straight edge elements for constant increments of arc length. To calculate the local normal rake angle at the serrated edge point, Local Rake Normal Correspondence used the normal vector on the rake face of the corresponding unserrated helical edge point at the same axial cross section. Uncut chip thickness h was computed under Local Oblique Cutting Correspondence published earlier. Stabler’s rule specified chip flow direction and Lazoglu mechanistic normal and friction force coefficients completed the straight edge cutting element forces. Local force contribution vectors summed over the tool engagement resulted in global tool forces. Number of straight edge element computations were minimized with the hypotheses (a) cutting predominantly occurs at the vicinity of serration tips and (b) only the stock surfaces swept by flutes immediately prior and following the current flute need be considered for h. Interaction of axial cross sections at a significant fraction of the serration wavelength was noted. Thus, treating tool axial subdivisions in relative isolation for h is untenable. However, local normal rake angle on the serrated edges followed earlier published trends. For a square endmill, sinusoidally serrated, cutting with low axial depth of cut ADOC, modeled forces showed idle periods and unbalance fluctuations during a full rotation, but not the data; modeled forces for an identical unserrated helical cutter were lower than the serrated. For the same endmill with circular serrations cutting at high ADOC, no idle periods or unbalance were predicted and the agreement with published data was better.

Outlier Detection of Functional Data Using Reproducing Kernel Hilbert Space

The problem of finding the pattern that deviates from other observation is termed as outlier. The detection of outlier is getting importance in research area nowadays due to the reason that the technique has been used in various mission critical applications such as military, health care, fault recovery, and many. The analysis of functional data and its depth function plays a crucial role in statistical model for detecting outlier. The depth values alone not enough for finding outliers, since all the low depth values not be an outlier. The main problem of using classical model is that it cannot cop up with the high dimensionality of the data This paper proposed a novel technique based on Reproducing Kernel Hilbert Space curve (RKHS) for detecting outliers in functional data. The proposed RKHS model is based on a special Hilbert space curve associated with a kernel so that it reproduces each function in the space to enhance the performance of data depth function. The proposed method uses distance weighted discrimination classification that avoids overfitting the model and provides better generalizability in high dimensions. The kernel depths perform better performances for detection of outlier in a number of artificial and real data sets.