Dual Curve Research Articles

Enhanced Salp Swarm Algorithm for Optimizing the Shape of Developable Bézier-Like Surfaces

The shape optimization of developable surfaces is a key and difficult technology in CAD/CAM. This paper studies the shape optimization of generalized developable Bézier-like surfaces using the improved salp swarm algorithm. First, aiming at the problems of slow convergence speed and low calculation accuracy, quickly falling into the optimal local solution for the salp swarm algorithm, and by adjusting the control parameters, the added Morlet wavelet mutation strategy to the individual position updates operation. An improved salp swarm algorithm is proposed, called FMSSA for short. Numerical results show that FMSSA performed the best among 18 (78%) comparisons with other improved SSAs in the CEC2005 test set. In addition, FMSSA serves the best among 16 (69%) comparisons with other algorithms. Therefore, the convergence speed and computational accuracy of FMSSA are improved, and the computational results are significantly better than other intelligent algorithms. Second, according to the duality principle of point and plane, the shape optimization problem of generalized developable Bézier-like surface is transformed into the minimization problem of arc length, energy, and curvature change rate of the dual curve, and three shape optimization models of the developable surface are established. Finally, shape optimization of expandable surfaces is mathematically an optimization problem that the swarm intelligence algorithm can efficiently handle, so FMSSA is used to solve these three optimization models. Numerical experiments show that FMSSA obtains optimal generalized developable Bézier-like surfaces in solving these three optimization models, demonstrating the superiority of the proposed FMSSA in effectively solving the shape optimization models in terms of precision and robustness.

Construction of WS2/Au-lipid drug delivery system for multiple combined therapy of tumor

The combination of chemo-photothermal therapy applied to tumor ablation, overcoming the shortcomings of traditional cancer treatment and has become a research hotspot. Tungsten disulfide nanosheets (WS2) are used as the core, and gold nanoparticles is incorporated into the surface layer of WS2 (WS2/Au) by the sodium citrate reduction method, and then gold-doped tungsten disulfide surface is coated by lipids (WS2/Au-lipids). The biocompatibility and stability of this functional nanocomposite has been strongly improved under physiological conditions. The excellent photothermal conversion characteristics of WS2/Au-lipid has been verified by the photothermal experiment both in vitro and in vivo. Furthermore, the DOX loaded by WS2/Au-lipid has a dual response release curve of pH and NIR. The results of in vitro photothermal and chemotherapy combined treatment shows that the survival rate of the cell of the combined treatment group is obviously lower than the single treatment group. Through the investigation of photothermal performance in vivo, it has been found that after modification, the photothermal performance of the compound has been improved, and the drug can be targeted for delivery to the tumor site, making it more concentrated in cancer cells. In addition, in vivo study confirms that the combined treatment group has a more obvious therapeutic effect on tumors than the single treatment group, and no systemic toxicity was shown. It is an excellent carrier in many aspects and has unlimited development and prospects in the field of multiple cancer treatments.

On an Explicit Characterization of Spherical Curves in Dual Lorentzian 3-SpaceD13

In this study, we developed a mathematical framework for studying spherical and null curves in dual Lorentzian 3-spaceD13. Considering the causal character of the dual curve, sufficient and necessary conditions for spacelike curves to be dual Lorentzian spherical were given. Also, new sufficient and necessary conditions for non-null curves to be hyperbolic dual spherical were presented. Then, an insight on curves with null-principal normal is reported.

A Dynamical Approach to Position Vector of Timelike Curve by Vectorial Momentum, Torque and Tangential Dual Curve

In this study, the position vector of a timelike curve wp is stated by a linear combination of its Serret Frenet frame with differentiable functions. The definition of tangential dual curve of the curve wp is stated by using these differentiable functions. Moreover, tangential torque curve of timelike curve wp is defined and investigated. New dynamically and physical results are stated depending on the torque of the timelike curve wp and the direction of the tangent vector component of the curve. Then, the position vector of a timelike W curve is again stated by differentiable functions. Therefore, solutions of differential equation of the position vector of timelike W curve with two different types depending on the values of curvature and torsion of timelike curve are obtained. By using the differentiable functions obtained as a result of these solutions, tangential dual and torque curve of the timelike W curve are obtained. Depending on the tangential dual and torque curve of the timelike W curve, results are given for two different cases separately.

Trajectory Planning and Optimization for Robotic Machining Based On Measured Point Cloud

Industrial robots are characterized by good flexibility and a large working space, and offer a new approach for the machining of large and complex parts with small machining allowances (extra material allowed for subsequent machining). Parts of this type (such as aircraft skin parts, wind turbine blades, etc.) are easily deformed due to their large scale and low stiffness. Therefore, these parts cannot be directly machined according to the designed model. A feasible method is to plan a robotic machining path by using the point clouds of parts after clamping from onsite measurement which contains inherent defects of measurement such as noise points and abrupt points. In this article, a novel method is proposed to plan and optimize a robotic machining path that meets the requirements of smoothness, dexterity, and stiffness based on the point cloud from onsite measurement. The dual nonuniform rational B-spline curves of the machining path points and tool axis points are generated at first. Next, an objective function of smoothness optimization is established to filter out the local mutation of the path by considering the constraints of both the deformation energy and the deviation. Then, the objective function of robot postures optimization is established to optimize dexterity and Cartesian stiffness of a robot during the machining process. To show the feasibility of the proposed method, simulation and experiments are carried out. It is proved that the proposed method can generate a smooth machining trajectory. The stability of joint rotation and the rigidity and dexterity of the robot are improved during the machining process.

The Dual Expression of Parallel Equidistant Ruled Surfaces in Euclidean 3-Space

In this study, we examine the dual expression of Valeontis’ concept of parallel p-equidistant ruled surfaces well known in Euclidean 3-space, according to the Study mapping. Furthermore, we show that the dual part of the dual angle on the unit dual sphere corresponds to the p-distance. We call these ruled surfaces we obtained “dual parallel equidistant ruled surfaces” and we briefly denote them with “DPERS”. Furthermore, we find the Blaschke vectors, the Blaschke invariants and the striction curves of these DPERS and we give the relationships between these elements. Moreover, we show the relationships between the Darboux screws, the instantaneous screw axes, the instantaneous dual Pfaff vectors and dual Steiner rotation vectors of these surfaces. Finally, we give an example, which we reinforce this article, and we explain all of these features with the figures on the example. Furthermore, we see that the corresponding dual curves on the dual unit sphere to these DPERS are such that one of them is symmetric with respect to the imaginary symmetry axis of the other.

Associated curves of a Frenet curve in the dual Lorentzian space

In this work, we firstly introduce notions of principal directed curves and principal donor curves which are associated curves of a Frenet curve in the dual Lorentzian space D 3 1 D13 . We give some relations between the curvature and the torsion of a dual principal directed curve and the curvature and the torsion of a dual principal donor curve. We show that the dual principal directed curve of a dual general helix is a plane curve and obtain the equation of dual general helix by using position vector of plane curve. Then we show that the principal donor curve of a circle in $\mathbb{D}^{2}$ or a hyperbola in $\mathbb{D}_{1}^{2}$ and the principal directed curve of a slant helix in $\mathbb{D}_{1}^{3}$ are a helix and general helix, respectively. We explain with an example for the second case. Finally, according to causal character of the principal donor curve of principal directed rectifying curve in $\mathbb{D}_{1}^{3}$, we show this curve to correspond to any timelike or spacelike ruled surface in Minkowski 3−space R 3 1 R13 .

A cross-cultural analysis of VR gaming psychological needs and motivations: a self-determination theory approach

ABSTRACT Purpose: Grounded in self-determination theory, this study examined whether the virtual reality gaming experience meets individuals’ psychological needs and develops their interest in VR sports games. Research methods: The authors conducted an exploratory study to examine video gamers’ motivation for playing VR games using interviews with 29 participants from the U.S. and Nigeria. Results and findings: The study found that most participants had two psychological needs met, namely competence and autonomy, while relatedness was not found. The participants demonstrated intrinsic motivation, as they expressed an interest to play VR games or to use a VR headset in the future. However, their current motivation status, in terms of their interest in purchasing a VR headset, can be qualified as identified regulation. Immersive experience was also found to be a major theme. Comparing Nigerian and U.S. participants, the Nigerians experienced dual learning curves. A few Nigerian participants indicated that they felt controlled by an external source in the game, while the U.S. participants were frustrated due to the limited controllable options available in the game. Some Nigerian participants also felt as if they were interacting with computer-driven characters. Implications: This study extends the research in this area and provides practical implications for various stakeholders involved in VR gaming technology.

Incidences between points and curves with almost two degrees of freedom

We study incidences between points and (constant-degree algebraic) curves in three dimensions, taken from a family C of curves that have almost two degrees of freedom, meaning that (i) every pair of curves of C intersect in O(1) points, (ii) for any pair of points p, q, there are only O(1) curves of C that pass through both points, and (iii) there exists a 6-variate real polynomial F of constant degree, so that a pair p, q of points admit a curve of C that passes through both of them if and only if F(p,q)=0. (As an example, the family of unit circles in R3 that pass through some fixed point is such a family.)We begin by studying two specific instances of this scenario. The first instance deals with the case of unit circles in R3 that pass through some fixed point (so called anchored unit circles). In the second case we consider tangencies between directed points and circles in the plane, where a directed point is a pair (p,u), where p is a point in the plane and u is a direction, and (p,u) is tangent to a circle γ if p∈γ and u is the direction of the tangent to γ at p. A lifting transformation due to Ellenberg et al. [3] maps these tangencies to incidences between points and curves (‘lifted circles’) in three dimensions. In both instances we have a family of curves in R3 with almost two degrees of freedom.We show that the number of incidences between m points and n anchored unit circles in R3, as well as the number of tangencies between m directed points and n arbitrary circles in the plane, is O(m3/5n3/5+m+n).We then derive a similar incidence bound, with a few additional terms, for more general families C of curves in R3 with almost two degrees of freedom, under a few additional natural assumptions.The proofs follow standard techniques, based on polynomial partitioning, but they face a critical novel issue involving the analysis of surfaces that are infinitely ruled by the respective family of curves, as well as of surfaces in a dual three-dimensional space that are infinitely ruled by the respective family of suitably defined dual curves. We either show that no such surfaces exist, or develop and adapt techniques for handling incidences on such surfaces.The general bound that we obtain isI(P,C)=O(m3/5n3/5+(m11/15n2/5+n8/9)δ1/3+m2/3n1/3π1/3+m+n), where π (resp., δ) is the maximal number of curves (resp., dual curves) that lie on a common surface that is infinitely ruled by the family of these curves.

Dual Spline Curve Interpolation Method for Continuous or Discontinuous Curve Representation

Dual Spline Curve Interpolation Method for Continuous or Discontinuous Curve Representation

The forward kinematics of dual rolling contact of one dual unit sphere on another with dual Darboux frame based equation

In this paper, we investigate the forward kinematics of dual rolling contact motion without sliding of one dual unit sphere $\tilde{S}_{m}^{2}$ on the fixed sphere $\tilde{S}_{f}^{2}$ along their dual spherical curves, which correspond to ruled surfaces generated by the straight lines in the real line space $\mathbb{E}^{3}$. We adopt a dual Darboux frame method to develop instantaneous kinematics of dual rolling motion. We obtain some new kinematic equations of rolling motion of the moving sphere $\tilde{S}_{m}^{2}$ with regards to dual unit vectors, dual rolling velocity, and dual geometric invariants. Namely, the dual translational velocity equation of an arbitrary dual point and the equation of the dual angular velocity on the moving sphere $\tilde{S}_{m}^{2}$ are derived. The equation represented by geometric invariants can be handily generalized to suit arbitrary dual spherical curve on $\tilde{S}_{m}^{2}$ and can be differentiated to any order.

On the role of dependence in sticky price and sticky information Phillips curve: Modelling and forecasting

Understanding the role of sticky price and sticky information for inflation dynamics is a key issue in economics. The literature has treated the two forms of stickiness as independent. This paper proposes a new dual stickiness Phillips curve based on dependence among the events of setting prices and updating information. Using US data over the period 1947Q1–2020Q1, the new model is scrutinized against a dual stickiness model without dependence, a pure sticky price model, and a pure sticky information model, through in- and out-of-sample analyses. The results show: (i) the new model outperforms the model without dependence in-sample; (ii) the dual stickiness models perform similarly out-of-sample; and (iii) the pure sticky models yield the worst forecasts. The results have some implications for policy makers and practitioners. A policy maker may consider the new model given its performance in- and out-of-sample, while a practitioner may prefer the model without dependence, given its lesser complexity and its competitive forecasting performance.

Central metal and ligand effects on oxygen electrocatalysis over 3d transition metal single-atom catalysts: A theoretical investigation

Metal single-atom catalysts (SACs) have recently emerged as promising alternatives to precious metal-based electrocatalysts for the oxygen reduction reaction (ORR) and oxygen evolution reaction (OER) under alkaline conditions. Currently, the role of the metal centre and local coordination environment on the ORR/OER activity of metal SACs remains unclear. Herein, we employed density functional theory (DFT) calculations to systematically study oxygen electrocatalysis on 60 different 3d transition metal SACs (scandium to zinc, supported on heteroatom-doped graphene supports), encompassing three symmetric nitrogen coordination configurations (i.e., M−Nn−C, n = 4, 3, 2) and three asymmetric coordination configurations (i.e., M−N3X−C, X = P, S, B). The calculations reveal the central metal and the coordinating atoms strongly influence oxygen electrocatalysis over 3d transition metal SACs, predominantly by tuning the adsorption free energy of adsorbed hydroxyl (ΔG*OH, a key descriptor of both ORR and OER activity). Dual limiting potential volcano curves were constructed for ORR and OER, with Ni-N2-C identified as the optimal synthetic target for bifunctional ORR/OER electrocatalysis, closely followed by Fe-N4-C, Co-N4-C, Co-N2-C, and Ni-N3P-C.

A special configuration of 12 conics and generalized Kummer surfaces

A generalized Kummer surface X obtained as the quotient of an abelian surface by a symplectic automorphism of order 3 contains a $$9\mathbf{A}_{2}$$ -configuration of $$(-2)$$ -curves. Such a configuration plays the role of the $$16\mathbf{A}_{1}$$ -configurations for usual Kummer surfaces. In this paper we construct 9 other such $$9\mathbf{A}_{2}$$ -configurations on the generalized Kummer surface associated to the double cover of the plane branched over the sextic dual curve of a cubic curve. The new $$9\mathbf{A}_{2}$$ -configurations are obtained by taking the pullback of a certain configuration of 12 conics which are in special position with respect to the branch curve, plus some singular quartic curves. We then construct some automorphisms of the K3 surface sending one configuration to another. We also give various models of X and of the generic fiber of its natural elliptic pencil.

The magnetic power on natural convection of NEPCM suspended in a porous annulus between a hexagonal-shaped cavity and dual curves

This paper discusses the magnetic powers on the thermosolutal convection of nano-encapsulated phase change material (NEPCM) inside a porous annulus by using the ISPH method. A novel annulus is constructed between an exterior hexagonal-shaped and inner shape of the dual curves having varying lengths (a&b). The heat/mass source is put in the left-top and right-bottom parts of a hexagonal-shaped and the other parts are adiabatic. The inner dual curves are maintained at Tc and Cc. The effects of the inner dual curve lengths (a=0.1−0.4&b=0.2−1), a fusion temperature (θf=0.05−0.95), Darcy parameter (Da=10−2−10−5), Hartmann number (Ha=0−100), solid volume fraction (ϕ=0.01−0.06), time parameter (τ=0−0.8), and Rayleigh number (Ra=103−106) on the contours of heat capacity, concentration, streamlines, temperature, and nanofluid velocity are researched. The results revealed that an increase in a fusion temperature shifts a phase change zone from the dual curves towards the heater source positions. The lengths of the inner dual curves are controlling the contours of the heat capacity, temperature, streamlines, and concentration in an annulus. Thus, the velocity's maximum boosts by 26.92% as length b shrinks from 1 to 0.2. Increasing Ra expands the contours of temperature, concentration, and heat capacity Cr across an annulus.

Exchange variations of generalized dual parallel curves and surfaces

Parallel curves (or offset curves) and parallel surfaces (or offset surfaces) have a big importance for CAD/CAM, robotics, cam design and many industrial applications, especially for mathematical modelling of cutting paths milling machines. Any vector space has a corresponding dual vector space that consists of all linear functions on vector space. Dual spaces are used in mathematics such as describing measures, distributions, and Hilbert spaces. Consequently, the dual space is an important concept in functional analysis. This paper proposes a novel definition of generalized and standard dual parallel curves and surfaces. Additionally, we give some properties of generalized dual parallel curves and surfaces using this novel definition. We also express the variation of the generalized dual parallel curves, the first and second variation of area change of the standard dual parallel surfaces and the first variation of area change of the generalized dual parallel surfaces.

Quaternionic and Dual Quaternionic Darboux Ruled Surfaces

In this paper, firstly the ruled surface drawn by the Darboux vector is expressed as a quaternion. Then, the spatial quaternionic definition of the striction curve is given and the integral invariants of the surface are calculated. Finally, the ruled surface which corresponds to a dual curve drawn by a dual Darboux vector is derived with the help of dual spatial quaternions and dual integral invariants of the ruled surface are obtained.

Curves of geodesic centers and Poncelet ellipses

In this paper, we consider smooth, closed, strictly convex curves contained inside the unit circle T and their associated curves of geodesic centers (or dual curves). We obtain a formula for the boundary and envelope of the union of the region bounded by the geodesic circles associated with our curves. This description includes a formula for the boundary of the hyperbolic convex hull of points identified by finite Blaschke products. We describe a setting and conditions under which one can produce an infinite chain of ellipses such that each ellipse is inscribed in a convex polygon that is itself inscribed in another polygon (a chain of Poncelet ellipses). We apply these results to operators that are compressions of the shift operator on model spaces and describe necessary and sufficient conditions for the numerical range to contain 0.

Singularities of the dual curve of a certain plane curve in positive characteristic

It is well known that the Gauss map for a complex plane curve is birational, whereas the Gauss map in positive characteristic is not always birational. Let $q$ be a power of a prime integer. We study a certain plane curve of degree $q^2 + q + 1$ for which the Gauss map is inseparable with inseparable degree $q$. As a special case, we show a relation between the dual curve of the Fermat curve of degree $q^2 + q + 1$ and the Ballico-Hefez curve.

Limits of dual curves via foliations

We develop a method to compute limits of dual plane curves in Zeuthen families of any kind. More precisely, we compute the limit 0-cycle of the ramification scheme of a general linear system on the generic fiber, only assumed geometrically reduced, of a Zeuthen family of any kind.