Quintic Curves Research Articles

Control point modifications that preserve the Pythagorean–hodograph nature of planar quintic curves

Although planar Pythagorean–hodograph (PH) curves are compatible with the standard Bernstein–Bézier representations, freely modifying the control points will compromise their PH nature. The present study focuses on identifying control point displacements that ensure a given planar PH curve remains a PH curve. In particular, for planar quintic PH curves r(t), t∈[0,1] it is shown that finitely-many simultaneous displacements of two control points yield modified quintic PH curves, identified as the solutions of quadratic and cubic equations. As a more practical approach, modification of PH quintics in canonical form with r(0)=0 and r(1)=1 by the displacement of a single interior control point is considered, with the remaining interior control points being used to minimize a measure of deviation from the original PH quintic. As illustrated by several examples, this approach provides an efficient and intuitive means of effecting reasonable shape modifications within the space of planar quintic PH curves.

Corrigendum to “Singular points of real quartic and quintic curves” by David A. Weinberg and Nicholas J. Willis [Tbilisi Mathematical Journal, Vol. 2, 2009, pp. 95-134

Corrigendum to “Singular points of real quartic and quintic curves” by David A. Weinberg and Nicholas J. Willis [Tbilisi Mathematical Journal, Vol. 2, 2009, pp. 95-134

Influence of the contraction section configuration and deflector layout density on the wind-snow flow characteristics inside the ice snow wind tunnel for railway vehicles

The significant accumulation of snow and ice in the bogie region severely compromises operational quality and poses a safety hazard to high-speed trains. To enhance testing capabilities for assessing ice and snow accumulation within bogie regions, this study conducts numerical investigations into the influence of contraction section configuration and deflector layout density on wind-snow flow quality within an ice and snow wind tunnel. The predictive accuracy of the numerical method is comprehensively validated against wind-snow wind tunnel tests, involving a comparison of airflow velocity in the test section and snow accumulation thickness on the channel's floor. The results demonstrate that the coupled numerical method of Improved Delayed Detached Eddy Simulation (IDDES) and Discrete Phase Model (DPM) is a reliable tool for analyzing wind-snow flow characteristics within an ice and snow wind tunnel. Furthermore, compared to the vickers curve and bicubic curve, quintic curves applied to the contraction section consistently compress and accelerate airflow. This contributes to a more uniform flow characterized by lower stream-wise fluctuation levels, pitch angles, and yaw angles in the test section, thereby significantly enhancing the overall flow quality of wind and snow. Thus, the quintic curves or smoother curves are recommended for the design of the contraction section in ice and snow wind tunnels. Moreover, increasing the deflector number leads to a noticeable improvement in flow quality inside the test section, while resulting in increased snow accretion on the windward surfaces of deflectors. Hence, a medium-level deflector density is suggested for designing ice and snow wind tunnels to achieve a better balance between flow quality and the additional snow accretion inside the channel.

Construction of planar quintic Pythagorean-hodograph curves by control-polygon constraints

In the construction and analysis of a planar Pythagorean–hodograph (PH) quintic curve r(t), t∈[0,1] using the complex representation, it is convenient to invoke a translation/rotation/scaling transformation so r(t) is in canonical form with r(0)=0, r(1)=1 and possesses just two complex degrees of freedom. By choosing two of the five control–polygon legs of a quintic PH curve as these free complex parameters, the remaining three control–polygon legs can be expressed in terms of them and the roots of a quadratic or quartic equation. Consequently, depending on the chosen two control–polygon legs, there exist either two or four distinct quintic PH curves that are consistent with them. A comprehensive analysis of all possible pairs of chosen control polygon legs is developed, and examples are provided to illustrate this control–polygon paradigm for the construction of planar quintic PH curves.

Explicit two-cover descent for genus 2 curves

Given a genus 2 curve C with a rational Weierstrass point defined over a number field, we construct a family of genus 5 curves that realize descent by maximal unramified abelian two-covers of C, and describe explicit models of the isogeny classes of their Jacobians as restrictions of scalars of elliptic curves. All the constructions of this paper are accompanied by explicit formulas and implemented in Magma and/or SageMath. We apply these algorithms in combination with elliptic Chabauty to a dataset of 7692 genus 2 quintic curves over $$\mathbb {Q}$$ of Mordell–Weil rank 2 or 3 whose sets of rational points have not previously been provably computed. We analyze how often this method succeeds in computing the set of rational points and what obstacles lead it to fail in some cases.

G1 Hermite interpolation method for spatial PH curves with PH planar projections

The research subject of this paper is the spatial Pythagorean hodograph (PH) curves whose projections to the horizontal plane are planar PH curves. Because of this geometric configuration, we name them PH curves over PH curves, or PHoPH curve. We investigate the algebraic structure of PHoPH curves and show that their hodographs are obtained by applying two squaring maps successively to quaternion generator polynomials. The simplest nontrivial PHoPH curves generated from linear quaternion generators are quintic curves, which have adequate degrees of freedom to solve the G1 Hermite interpolation problem. From the algebraic structure, we can derive a system of nonlinear equation for G1 interpolation, which is addressable by numerical methods. We also suggest the choice of initial values for the numerical method. The solvability is not guaranteed for arbitrary G1 data in general, however, we show the feasibility of the system for the G1 data taken from a small segment of reference curves without inflection points using extensive Monte-Carlo simulation. We also present a few illustrative examples of PHoPH spline curves that approximate the given reference curves.

A novel strategy for comprehensive optimization of structural and operational parameters in a supersonic separator using computational fluid dynamics modeling

In this study, the effects of several structural and operational parameters affecting the separation efficiency of supersonic separators were investigated by numerical methods. Different turbulence models were used and their accuracies were evaluated. Based on the error analysis, the V2-f turbulence model was more accurate for describing the high swirling turbulent flow than other investigated turbulence models. Therefore, the V2-f turbulence model and particle tracing model were selected to optimize the structure of the convergence part, the diffuser, the drainage port, and the swirler. The cooling performance of three line-type in the convergent section were calculated. The simulation results demonstrated that the convergent section designed by the Witoszynski curve had higher cooling depth compared to the Bi-cubic and Quintic curves. Furthermore, the expansion angle of 2° resulted in the highest stability of fluid flow and therefore was selected in the design of the diffuser. The effect of incorporating the swirler and its structure on the separation performance of supersonic separator was also studied. Three different swirler types, including axial, wall-mounted, and helical, were investigated. It was observed that installing the swirler significantly improved the separation efficiency of the supersonic separator. In addition, the simulation results demonstrated that the separation efficiency was higher for the axial swirler compared to the wall-mounted and helical swirlers. Therefore, for the improved nozzle, the swirling flow was generated by the axial swirler. The optimized axial swirler was constructed from 12 arced vanes each of which had a swirl angle of 40°. For the optimized structure, the effects of operating parameters such as inlet temperature, pressure recovery ratio, density, and droplet size was also investigated. It was concluded that increasing the droplet size and density significantly improved the separation efficiency of the supersonic separator. For hydrocarbon droplets, the separation efficiency improved from 4.6 to 76.7% upon increasing the droplet size from 0.1 to 2 µm.

Roll crown control capacity of sextic CVC work roll curves in plate rolling process

The shape curve of continuous variable crown (CVC) roll applied in plate rolling process is usually adopted as cubic curves or quintic curves, but the coupling between quadratic crown and quartic crown of conventional CVC roll shape curve could not work well and failed to meet the product quality requirements in plate rolling. To get better control effect on the composite wave shape, a sextic CVC roll shape curve was proposed. The theoretical calculation formula of the roll shape parameters was deduced by using the decomposition and superposition principle. Based on the actual production data of plate mill, sextic CVC work roll parameters were determined through the analysis of the convexity control ability of the curve. The application results showed that a sextic CVC roll shape curve can greatly improve the control ability of plate crown. The control ability of quadratic crown improved by 5% and that of quartic crown improved by 20%, respectively; further, the peak contact pressure of quintic CVC roll decreased from 2076 to 1744 MPa, and the difference between the maximum and the minimum of roll pressure was 184 MPa. The control ability of plate crown was controlled from − 67 to 63 μm with plate width equaling to 2400 mm. The deviation of measured value of plate crown and calculated value of plate crown was less than ± 4%.

SCR-Normalize: A novel trajectory planning method based on explicit quintic polynomial curves

This paper presents a framework for generating explicit quintic polynomial curves as the trajectory of autonomous vehicles. The method is called SCR-Normalize and is founded on two novel ideas. The first concept is to rotate the coordinate reference, regarding the boundary conditions, in order to reach a special coordinate reference called Secondary Coordinate Reference (SCR). In the SCR, the explicit quintic polynomial curve has short length and low curvature values (i.e. the curve is not wiggly). The second concept is to normalize the trajectory in order to speed up the framework. Two kinds of problems are considered to be solved in this paper: (a) generating a length optimal trajectory for arbitrary boundary conditions subject to a curvature constraint; and (b) path smoothing. For case (a), for the lane change manoeuvre, the problem is solved explicitly. In addition, the familiar expression for the optimal interval of the lane change manoeuvre is analytically proved for the first time. For arbitrary swerving manoeuvres, two algorithms are presented and their performance is compared with each other and with two other algorithms. Similarly, for case (b), an algorithm is presented and its performance is compared with three other methods. Evaluating the algorithms in the examples, and comparing the results with other methods illustrates the efficiency of the SCR-Normalize framework to generate optimal trajectories in real time.

A new selection scheme for spatial Pythagorean hodograph quintic Hermite interpolants

For given C1 Hermite data, there exists a two-parameter family of Pythagorean hodograph (PH) quintic curves which interpolate the data (two end-points and end-derivatives) as observed by Farouki et al. (2002b). As “good” candidate curves for a selection problem, we propose a special type of PH quintic interpolating curves called extremal interpolants and prove that the extremal interpolants preserve planarity, i.e., they are planar curves if the data are planar. Since there are only four distinct extremal interpolants, the selection problem, when only considering extremal interpolants as possible candidates, is reduced to picking one curve from finite candidates.Due to the preservation of planarity, extremal interpolants coincide with one of p0,0(t),p0,π(t),pπ,0(t),pπ,π(t) if the data are planar, where pϕ0,ϕ2(t) denotes the parametrization proposed by Šír and Jüttler (2005). However, any of the four extremal interpolants is generically not identical to the interpolants for non-planar data, and empirical results suggest that being compared with the unique cubic interpolant, the best curve is one of extremal interpolants among all extremal interpolants and p0,0(t),p0,π(t),pπ,0(t),pπ,π(t).

Trajectory Planning for Rotational Chain Shell Magazine

To ensure the stability and reliability of the rotational chain shell magazine moving system, the trajectory planning such as cubic curve, quintic curve, T-shaped velocity curve and S-shaped velocity curve were considered in this paper. The effectiveness and feasibility of each trajectory planning method for the control system of the rotational chain shell magazine were verified by the combination simulation technology of MSC. ADAMS and MATLAB. The simulation results show that the S-shaped velocity curve satisfies the requirement of the smoothness and continuity of the trajectory of the magazine. It ensures the continuity of angular velocity and acceleration, and reduces the buffeting and impact effectively that other curves can’t satisfy. It is of great significance for the stability and reliability of the rotational chain shell magazine.

Deformation Classification of Real Non-singular Cubic Threefolds with a Marked Line

We prove that the space of pairs (X, l) formed by a real non-singular cubic hypersurface $$X\subset P^4$$ with a real line $$l\subset X$$ has 18 connected components and give for them several quite explicit interpretations. The first one relates these components to the orbits of the monodromy action on the set of connected components of the Fano surface $$F_{\mathbb R}(X)$$ formed by real lines on X. For another interpretation we associate with each of the 18 components a well defined real deformation class of real non-singular plane quintic curves and show that this deformation class together with the real deformation class of X characterizes completely the component.

Covers of stacky curves and limits of plane quintics

We construct a well-behaved compactification of the space of finite covers of a stacky curve using admissible cover degenerations. Using our construction, we compactify the space of tetragonal curves on Hirzebruch surfaces. As an application, we explicitly describe the boundary divisors of the closure inM6 of the locus of smooth plane quintic curves.

A note on Pythagorean hodograph quartic spiral

By using the geometric constraints on the control polygon of a Pythagorean hodograph (PH) quartic curve, we propose a sufficient condition for this curve to have monotone curvature and provide the detailed proof. Based on the results, we discuss the construction of spiral PH quartic curves between two given points and formulate the transition curve of a G2 contact between two circles with one circle inside another circle. In particular, we deduce an attainable range of the distance between the centers of the two circles and summarize the algorithm for implementation. Compared with the construction of a PH quintic curve, the complexity of the solution of the equation for obtaining the transition curves is reduced.

Construction of a class of quintic curves with rational offsets

In this paper, a method for constructing a class of quintic curves with rational offsets is presented. Although this class of planar curves does not include PH curves, such curves are widely applied in CAD because they have rational offsets. A complex variate model is employed to deduce the proposed method. The quintic OR curves are first classified into two classes according to the different factorization of their hodographs, and are then discussed. With the given $C^1$ Hermite data, the curves are determined by specifying a real parameter. This real parameter can be used to adjust the shape of the constructed curves, and affects the parameter values of the cusps.

Experimental investigation on thermal-hydraulic characteristics of endothermic hydrocarbon fuel in 1 mm and 2 mm diameter mini-channels

The thermal-hydraulic characteristics of endothermic hydrocarbon fuel were experimentally investigated in circular mini-channels with inner diameters of 1.0 and 2.0mm and heated length of 800.0mm. The effects of heat flux ranging from (400 to 1750)kW·m−2, pressure from (1.0 to 6.0)MPa, and inner diameter were studied. We found that the thermal-hydraulic characteristics of endothermic fuel could be described by a quintic curve. Two positive slope regions and two negative slope regions were experimentally obtained. The boiling or pseudo-boiling, and fuel pyrolysis which led to the rapid density reduction caused two negative slope regions in the curve. The negative slope region at fuel pyrolysis conditions always existed at the tested subcritical and supercritical pressures. The effect of heat flux on the thermal-hydraulic characteristics was not significant. The transition points were fixed and connected to the zeotropic process or pseudo-boiling temperature and pyrolysis reactions regardless of heat fluxes. The multi-valued characteristics can be reduced in small channels, as the magnitude of pressure drop variation was much larger in big channels. The study on thermal-hydraulic characteristics of endothermic fuel starts a new direction in fluid dynamics considering the chemical reaction.

Projection of a nonsingular plane quintic curve and the dihedral group of order eight

Let C be a nonsingular plane quintic curve over the complex number field \mathbb{C} , and let \pi_P\colon C \rightarrow \mathbb{P}^1 be a projection from P \in C . Let L_P be the Galois closure of the field extension \mathbb{C}(C)/\mathbb{C}(\mathbb{P}^1) induced by \pi_P , where \mathbb{C}(C) and \mathbb{C}(\mathbb{P}^1) are the rational function fields of C and \mathbb{P}^1 , respectively. We call the point P a D_4 -point if the Galois group of L_P/\mathbb{C}(\mathbb{P}^1) is isomorphic to the dihedral group D_4 of order eight. In this paper, we prove that the number of D_4 -points for C equals 0 , 1 , 3 , 5 , or 15 , and show that the curve with 15 D_4 -points is projectively equivalent to the Fermat quintic curve.

On the distribution of Weierstrass points on Gorenstein quintic curves

Abstract This paper is concerned with developing a technique to compute in a very precise way the distribution of Weierstrass points on the members of any 1-parameter family Ca, a ∈ C , of Gorenstein quintic curves with respect to the dualizing sheaf K C a . The nicest feature of the procedure is that it gives a way to produce examples of existence of Weierstrass points with prescribed special gap sequences, by looking at plane curves or, more generally, to subcanonical curves embedded in some higher dimensional projective space.

A Composite COG Trajectory Planning Method for the Quadruped Robot Walking on Rough Terrain

The COG trajectory planning method is the primary concern in the gait planning for quadruped robot, especially when the quadruped robot travelling on rough terrain. In this paper, we focus on the scenario where the quadruped robot walking on the rough terrain with the static walking gait. We present a smooth COG trajectory generator that the COG smooth trajectory characterized by continuous velocity and acceleration profiles can be generated automatically according to the current foot placement of the robot. A composite COG trajectory composed of quintic curves and straight lines is proposed in this paper. When the robot swings a leg forward, the COG trajectory is desired to be a straight line to eliminate the influence of acceleration of the body on the stability of the robot. However, the COG trajectory is desired to be a quintic curve when the robot is in the four legs support stage to guarantee the continuity of the COG trajectory. Moreover, the walk efficiency and the ability to keep the stability of robot are considered in this paper. Base on the COG trajectory generate method and the gait planning provided in this paper, the robot can walk through the rough terrain as soon as possible in the condition that the stability margin of the robot no less than the minimum stability margin. Via simulation the performance of the proposed COG trajectory is verified.

Virtual Poincaré polynomial of the space of stable pairs supported on quintic curves

Let $\mathbf{M}^{\alpha}(d,\chi)$ be the moduli space of $\alpha$-stable pairs $(s,F)$ on the projective plane $\mathbb{P}^2$ with Hilbert polynomial $\chi(F(m))=dm+\chi$. For sufficiently large $\alpha$ (denoted by $\infty$), it is well known that the moduli space is isomorphic to the relative Hilbert scheme of points over the universal degree $d$ plane curves. For the general $(d,\chi)$, the relative Hilbert scheme does not have a bundle structure over the Hilbert scheme of points. In this paper, as the first non trivial such a case, we study the wall crossing of the $\alpha$-stable pairs space when $(d,\chi)=(5,2)$. As a direct corollary, by combining with Bridgeland wall crossing of the moduli space of stable sheaves, we compute the virtual Poincar\'e polynomial of $\mathbf{M}^{\infty}(5,2)$.