Space Curves Research Articles

Elements of the parabola in the three-dimensional space and applications in teaching through a project based on reflection of light

This work, structured in two main parts, is devoted to the parabola topic. In the first, the elements of a parabola are reviewed by linking the definition of intersection of a plane and a cone with a locus of the plane. The necessary transformations for the calculation of the elements as a curve in space are pointed out as well as historical notes and properties of light are included. In the second part, the applications in teaching at a high school level or university courses are explained. A project based on the reflection of light is proposed that seeks to connect various subjects in line with the new educational paradigm of development of key competencies, joining different fields of knowledge. For university courses, applications of linear algebra are obtained in the establishment of relationships between analytical geometry in the rotation and translation of planes, and the dihedral system that is taught in technical drawing subjects. As a conclusion, it is obtained that the teaching of the parabola from different approaches allows a complete learning of diverse fields of knowledge, even of different topics of mathematics, an essential factor in the development of thought at any educational level.

Irreducibility of periodic curves in cubic polynomial moduli space

AbstractIn the moduli space of complex cubic polynomials with a marked critical point, given any , we prove that the loci formed by polynomials with the marked critical point periodic of period is an irreducible curve. Thus, answering a question posed by Milnor in the 1990s.

Some Aspects of Rectifying Curves on Regular Surfaces Under Different Transformations

An essential space curve in the study of differential geometry is the rectifying curve. In this paper, we studied the adequate requirement for a rectifying curve under the isometry of the surfaces. The normal components of the rectifying curves are also studied, and it is investigated that for rectifying curves, the Christoffel symbols and the normal components along the surface normal are invariant under the isometric transformation. Moreover, we also studied some properties for the first fundamental form of the surfaces.

Differential Geometry of Space Curves: Forgotten Chapters

Differential Geometry of Space Curves: Forgotten Chapters

The formation of solitary wave solutions and their propagation for Kuralay equation

In this paper, the main motive is to mathematical explore the Kuralay equation, which find applications in various fields such as ferromagnetic materials, nonlinear optics, and optical fibers. The objective of this study is to investigate different types of soliton solutions and analyze the integrable motion of induced space curves. This article appropriates the traveling wave transformation allowing the partial differential equation to be changed into an ordinary differential equation. To establish these soliton solutions, the study employs the new auxiliary equation method. As an outcome, a numerous types of soliton solutions like, Periodic pattern with anti-peaked crests and anti-troughs, singular solution, mixed complex solitary shock solution, mixed singular solution, mixed shock singular solution, mixed trigonometric solution, mixed periodic, periodic solution and mixed hyperbolic solution obtain via Mathematica. In order to visualize the graphical propagation of the obtained soliton solutions, 3D, 2D, and contour graphics are generated by choosing appropriate parametric values. The impact of parameter w is also graphically displayed on the propagation of solitons.

Designing dynamically corrected gates robust to multiple noise sources using geometric space curves

Designing dynamically corrected gates robust to multiple noise sources using geometric space curves

Piecewise polynomial approximation of spatial curvilinear profiles using the Hough transform

Given a curvilinear profile P represented as a set of points in the space R3 and four families of low-degree polynomial curves that respectively depend on the parameters in the space R4, our goal is to identify the piecewise space polynomial curve best fitting the profile P. We use a parametric representation of the space curves and subdivide the profile into smaller portions that can be fitted with regular curves. We provide theoretical guarantees to the existence of such an approximation and an algorithm for the profile approximation. We take advantage of the implicit function theorem to locally project a space curve on at most two planes and to locally recognise it with a low-degree polynomial curve obtained by applying the Hough transform. Finally, we recombine the curve expressions on the two planes backwards in the space R3. The outcome of the algorithm is thus a piecewise polynomial curve approximating the profile. We validate our approach to approximate curvilinear profiles extracted from 3D point clouds representing real objects and to simplify and resample point clouds.

Hypergeometric viable models in f(R) gravity

A cosmologically viable hypergeometric model within the framework of the modified gravity theory f(R) has been derived based on the requirements of asymptotic behavior towards ΛCDM, the presence of an inflection point in the f(R) curve, and the viability conditions dictated by the phase space curves (m, r), where m and r denote characteristic functions of the model. To examine the constraints associated with these viability criteria, the models were expressed in terms of a dimensionless variable, namely R → x and f(R) → y(x) = x + h(x) + λ, where h(x) represents the deviation of the model from General Relativity. By employing the geometric properties imposed by the inflection point, differential equations were formulated to establish the relationship between and h″(x). The resulting solutions yielded models of the Starobinsky (2007) and Hu-Sawicki types. However, it was subsequently discovered that these differential equations correspond to specific cases of a hypergeometric differential equation, indicating that these models can be derived from a more general hypergeometric model. The parameter domains of this model were thoroughly analyzed to ensure its viability.

A Surface Family with a Mutual Geodesic Curve in Galilean 3-Space G3

This article gives an approach for establishing a surface family with a mutual geodesic curve in Galilean 3-space G3. Given a smooth space curve, we derive the sufficient and necessary condition for the given curve to be geodesic on it. Furthermore, we resolve the conditions when the surface is a ruled surface. Consequently, its ability to develop with the mutual geodesic of the elements of the surface family is researched. Meanwhile, we explain this approach by submitting considerable examples.

Jacobian-free variational method for computing connecting orbits in nonlinear dynamical systems.

One approach for describing spatiotemporal chaos is to study the unstable invariant sets embedded in the chaotic attractor of the system. While equilibria, periodic orbits, and invariant tori can be computed using existing methods, the numerical identification of heteroclinic and homoclinic connections between them remains challenging. We propose a robust matrix-free variational method for computing connecting orbits between equilibrium solutions. Instead of a common shooting-based approach, we view the identification of a connecting orbit as a minimization problem in the space of smooth curves in the state space that connect the two equilibria. In this approach, the deviation of a connecting curve from an integral curve of the vector field is penalized by a non-negative cost function. Minimization of the cost function deforms a trial curve until, at a global minimum, a connecting orbit is obtained. The method has no limitation on the dimension of the unstable manifold at the origin equilibrium and does not suffer from exponential error amplification associated with time-marching a chaotic system. Owing to adjoint-based minimization techniques, no Jacobian matrices need to be constructed. Therefore, the memory requirement scales linearly with the size of the problem, allowing the method to be applied to high-dimensional dynamical systems. The robustness of the method is demonstrated for the one-dimensional Kuramoto-Sivashinsky equation.

Sarkisov links with centre space curves on smooth cubic surfaces

Sarkisov links with centre space curves on smooth cubic surfaces

A Wavelet Basis ANN and 5-Class Decision Factor AI Algorithm

The accuracy and reliability of continuous space curve estimation is the key to global exploration. An improved artificial intelligence algorithm is proposed for continuous space analysis. First, a wavelet basis ANN algorithm is proposed to determine the discretization strategy in continuous space. The hidden layer node transfer function in a BP neural network is replaced by a wavelet basis function, and the modified BP neural network is composed of a wavelet neural network. Second, an improved wolf algorithm is established. The core wolf system ensures the precision of the whole exploration task. Finally, main and auxiliary double cores and a five-class decision factor are used to establish a population classification model to solve the convergence of the algorithm.

On the harmonic evolute of time-like Hasimoto surfaces in Lorentz–Minkowski space

The movement of a thin vortex in a thin viscous fluid by the motion of a curve propagating in Lorentz–Minkowski space [Formula: see text] is described by the vortex filament or smoke ring equation and can be viewed as a dynamical system on the space curves in [Formula: see text]. This paper investigates the harmonic evolute surfaces of time-like Hasimoto surfaces in [Formula: see text]. Also, we discuss the geometric properties of these surfaces, namely, we obtain the Gaussian and mean curvatures of the first and second fundamental forms. As a verification, we construct a concrete example for the meant surfaces to demonstrate our theoretical results.

A nonlinear transformation between space curves defined by curvature-torsion relations in 3-dimensional Euclidean space

In this paper, we define a nonlinear transformation between space curves which preserves the ratio of $\tau/\kappa$ of the given curve in 3−dimensional Euclidean space $E^3$. We investigate invariant and associated curves of this transformation by the help of curvature and torsion functions of the base curve. Moreover, we define a new curve (family) so-called quasi-slant helix, and we obtain some characterizations in terms of the curvatures of this curve. Finally, we examine some curves in the kinematics, and give the pictures of some special curves and their images with respect to the transformation.

Curves in quantum state space, geometric phases, and the brachistophase

Given a curve in quantum spin state space, we inquire what is the relation between its geometry and the geometric phase accumulated along it. Motivated by Mukunda and Simon’s result that geodesics (in the standard Fubini-Study metric) do not accumulate geometric phase, we find a general expression for the derivatives (of various orders) of the geometric phase in terms of the covariant derivatives of the curve. As an application of our results, we put forward the brachistophase problem: given a quantum state, find the (appropriately normalized) Hamiltonian that maximizes the accumulated geometric phase after time τ—we find an analytical solution for all spin values, valid for small τ. For example, the optimal evolution of a spin coherent state consists of a single Majorana star separating from the rest and tracing out a circle on the Majorana sphere.

Singularity properties of timelike circular surfaces in Minkowski 3-space

The approach of the study is on singularity properties of timelike circular surfaces in Minkowski 3-space. A timelike circular surface is a one-parameter set of Lorentzian circles with stationary radius directing a non-null space curve, which acts as the spine curve, and it has symmetrical properties. In this study, we addressed timelike circular surfaces and gained their geometric and singularity properties such as Gaussian and mean curvatures, comparable with those of ruled surfaces. Consequently, we presented timelike roller coaster surfaces as a special class of timelike circular surfaces. Then, the conditions for timelike roller coaster surfaces to be flat or minimal surfaces are obtained. Meanwhile, we supported the results of the approach with some examples.

A smooth compactification of the space of genus two curves in projective space: via logarithmic geometry and Gorenstein curves

A smooth compactification of the space of genus two curves in projective space: via logarithmic geometry and Gorenstein curves

Shearless curve breakup in the biquadratic nontwist map

Nontwist area-preserving maps violate the twist condition along shearless invariant curves, which act as transport barriers in phase space. Recently, some plasma models have presented multiple shearless curves in phase space and these curves can break up independently. In this paper, we describe the different shearless curve breakup scenarios of the so-called biquadratic nontwist map, a recently proposed area-preserving map derived from a plasma model, that captures the essential behavior of systems with multiple shearless curves. Three different scenarios are found and their dependence on the system parameters is analyzed. The results indicate a relation between shearless curve breakup and periodic orbit reconnection-collision sequences. In addition, even after a shearless curve breakup, the remaining curves inhibit global transport.

Duality for certain multi-Frobenius nonclassical curves in higher dimensional spaces

We show how a type of multi-Frobenius nonclassicality of a curve defined over a finite field Fq of characteristic p reflects on the geometry of its strict dual curve. In particular, in such cases we may describe all the possible intersection multiplicities of its strict dual curve with the linear system of hyperplanes. Among other consequence, using a result by Homma, we are able to construct nonreflexive space curves such that their tangent surfaces are nonreflexive as well, and the image of a generic point by a Frobenius map is in its osculating hyperplane. We also obtain generalizations and improvements of some known results of the literature.

Bi-Lipschitz Characterization of Space Curves

Bi-Lipschitz Characterization of Space Curves