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A smooth ruled surface in 4‐space has only parabolic points or inflection points of the real type. We show, by means of contact with transverse planes, that at a parabolic point, there exist two tangent directions determining two planes along which the parallel projection exhibits A$\mathcal {A}$‐singularities of type butterfly or worse. In particular, such parabolic points can be classified as butterfly hyperbolic, parabolic, or elliptic points depending on the value of the discriminant of a binary differential equation (BDE). Also, whenever such discriminant is positive, we ensure that the integral curves of these directions form a pair of foliations on the ruled surface. Moreover, the set of points that nullify the discriminant is a regular curve transverse to the regular curve formed by inflection points of the real type. Finally, using a particular projective transformation, we obtain a simple parametrization of the ruled surface such that the moduli of its 5‐jet identify a butterfly hyperbolic/parabolic/elliptic point, as well as we get the stable configurations of the solutions of BDE in the discriminant curve. [ABSTRACT FROM AUTHOR]

We prove that the modular component , constructed in the Main Theorem in Fania and Flamini (Adv Math 436:109409, 2024. https://doi.org/10.1016/j.aim.2023.109409), of Ulrich vector bundles of rank r and given Chern classes, on suitable threefold scrolls X e over Hirzebruch surfaces F e ≥ 0 , which arise as tautological embeddings of projectivization of very-ample vector bundles on F e , is generically smooth, irreducible and unirational. A stronger result holds for the suitable associated moduli space of vector bundles of rank r and given Chern classes on F e , Ulrich w.r.t. the very ample polarization which turns out to be generically smooth, irreducible and unirational. [ABSTRACT FROM AUTHOR]

We study deformed Hermitian Yang–Mills (dHYM) connections on ruled surfaces explicitly, using the momentum construction. As a main application, we provide many new examples of dHYM connections coupled to a variable background Kähler metric. These are solutions of the moment map partial differential equations given by the Hamiltonian action of the extended gauge group, coupling the dHYM equation to the scalar curvature of the background. The large radius limit of these coupled equations is the Kähler–Yang–Mills system of Álvarez-Cónsul, Garcia-Fernandez, and García-Prada, and in this limit, our solutions converge smoothly to those constructed by Keller and Tønnesen-Friedman. We also discuss other aspects of our examples including conical singularities, realization as B-branes, the small radius limit, and canonical representatives of complexified Kähler classes. [ABSTRACT FROM AUTHOR]

In this paper, our aim is to research the change in the magnetic field using the curvature theory of ruled surfaces. For this, we examine magnetic curves of the curve whose existence is guaranteed from the derivative formulas of the rotation frame. The Killing vector fields and Lorentz forces of these magnetic curves are calculated. Additionally, correlations between curvatures are obtained. Flux ruled surfaces are obtained and the conditions for developable are achieved. Finally, examples of the magnetic flux developable ruled surfaces are given. [ABSTRACT FROM AUTHOR]

In this paper, we consider curves on a cone that pass through the vertex and are also triple covers of the base of the cone, which is a general smooth curve of genus γ and degree e in P e - γ . Using the free resolution of the ideal of such a curve found by Catalisano and Gimigliano, and a technique concerning deformations of curves introduced by Ciliberto, we show that the deformations of such curves remain on cones over a deformation of the base curve. This allows us to prove that for γ ≥ 3 and e ≥ 4 γ + 5 , there exists a non-reduced component H of the Hilbert scheme of smooth curves of genus 3 e + 3 γ and degree 3 e + 1 in P e - γ + 1 . We show that dim T [ X ] H = dim H + 1 = (e - γ + 1) 2 + 7 e + 5 for a general point [ X ] ∈ H . [ABSTRACT FROM AUTHOR]

Ruled surfaces are considered one of the significant aspects of differential geometry. These surfaces are formed by the motion of a straight line called a generator, and every curve that intersects all the generators is called a directrix. In the present research paper, we explore a family of ruled surfaces constructed from circular helices (W-curve) using the Frenet frame in the Euclidean space E 3 . We derive the explicit formulas for the second mean curvature and second Gaussian curvature. We present some ruled surfaces, and we describe their properties. In addition, we determine the sufficient conditions for these surfaces to be minimal, flat, II-minimal, and II-flat. Also, we obtain sufficient conditions for the base curve for these ruled surfaces to be a geodesic curve, an asymptotic line, and a principal line. Furthermore, we present an application for a ruled surface whose base curve is a circular helix, we compute some quantities for this surface such as the mean curvature and Gaussian curvatures and we plot the ruled surface with its base curve, and at symmetric points and along a symmetry axis. [ABSTRACT FROM AUTHOR]

Conformable fractional calculus is a relatively new branch of mathematics that seeks to extend traditional calculus to include non-integer order derivatives and integrals. This new form of calculus allows for a more precise description of physical phenomena, such as those related to fractal geometry and chaos theory. It also offers new tools for solving problems in physics, engineering, finance, and other areas. By using conformable fractional calculus, researchers can gain insight into the behavior of systems that would otherwise be difficult or impossible to analyze. In this article, we will discuss the fundamentals of conformable fractional calculus and its applications in various fields. For this purpose ruled surfaces are studied with conformable fractional calculus. The ruled surface is rearranged concerning the conformable surface definition and geometric properties are investigated. [ABSTRACT FROM AUTHOR]

We show that a length-minimizing disk inherits the upper curvature bound of the target. As a consequence we prove that harmonic discs and ruled discs inherit the upper curvature bound from the ambient space. [ABSTRACT FROM AUTHOR]

In this paper, we define three types of 2‐ruled hypersurfaces in the Minkowski 4‐space 피14. We obtain Gaussian and mean curvatures of the 2‐ruled hypersurfaces of type‐1 and type‐2 and some characterizations about its minimality. We also deal with the first Laplace–Beltrami operators of these types of 2‐ruled hypersurfaces in 피14. Moreover, the importance of this paper is the definition of these surfaces by using the octonions in 피14. Thus, this is a new idea and makes the paper original. We give an example of 2‐ruled hypersurface constructed by octonion, and we visualize the projections of the images with MAPLE program. Furthermore, the optical fiber can be defined as a one‐dimensional object embedded in the four‐dimensional Minkowski space 피14. Thus, as a discussion, we investigate the geometric evolution of a linearly polarized light wave along an optical fiber by means of the 2‐ruled hypersurfaces in a four‐dimensional Minkowski space. [ABSTRACT FROM AUTHOR]

The spherical indicatrix of a ruled surface is the image curve of its rulings on the sphere S 2 . In this paper, we present Lorentz forces and magnetic curves produced by the Frenet frame { T e , N e , B e } of the spherical indicatrix of a ruled surface in a magnetic field. We calculate magnetic vector fields of magnetic curves for T e , N e , B e . Furthermore, we define magnetic flux surfaces constructed by magnetic vector fields along magnetic spherical indicatrix. We obtain developability conditions for these surfaces. Finally, we give some examples to show magnetic flux surfaces. [ABSTRACT FROM AUTHOR]

In this study, we take the striction line which only exists on a ruled surface as an optical fiber. We examine Berry phase model of an optical fiber with a geodesic Frenet frame. We find the direction of the polarized light concerning the plane in which the electric field Ê is located. Also, the electromagnetic trajectories of optical fiber are obtained. Finally, we give the electromagnetic flux ruled surfaces formed by the Killing vector fields of the EM-trajectories of the optical fiber and numerical examples. [ABSTRACT FROM AUTHOR]

This study begins with the construction of type- Π Smarandache ruled surfaces, whose base curves are Smarandache curves derived by rotation-minimizing Darboux frame vectors of the curve in E 3 . The direction vectors of these surfaces are unit vectors that convert Smarandache curves. The Gaussian and mean curvatures of the generated ruled surfaces are then separately calculated, and the surfaces are required to be minimal or developable. We report our main conclusions in terms of the angle between normal vectors and the relationship between normal curvature and geodesic curvature. For every surface, examples are provided, and the graphs of these surfaces are produced. [ABSTRACT FROM AUTHOR]

In this study, we first define the Smarandache curves derived from the Frenet vectors and the Darboux vector of any curve. Then, we construct new ruled surfaces along these Smarandache curves with the direction vectors obtained from the Frenet vectors and the Darboux vector, and give the equations of these surfaces. In addition, we calculate the Gaussian and mean curvatures of these surfaces separately and present the conditions to be minimal and developable for these surfaces. Finally, as an example, we obtain ruled surfaces whose base curves are Viviani's curves and plot the graphics of these surfaces. [ABSTRACT FROM AUTHOR]

In this paper, we study the singularities on a non-developable ruled surface according to Blaschke's frame in the Euclidean 3-space. Additionally, we prove that singular points occur on this kind of ruled surface and use the singularity theory technique to examine these singularities. Finally, we construct an example to confirm and demonstrate our primary finding.

Ruled surfaces are considered one of the significant aspects of differential geometry. These surfaces are formed by the motion of a straight line called a generator, and every curve that intersects all the generators is called a directrix. In the present research paper, we explore a family of ruled surfaces constructed from circular helices (W-curve) using the Frenet frame in the Euclidean space E3. We derive the explicit formulas for the second mean curvature and second Gaussian curvature. We present some ruled surfaces, and we describe their properties. In addition, we determine the sufficient conditions for these surfaces to be minimal, flat, II-minimal, and II-flat. Also, we obtain sufficient conditions for the base curve for these ruled surfaces to be a geodesic curve, an asymptotic line, and a principal line. Furthermore, we present an application for a ruled surface whose base curve is a circular helix, we compute some quantities for this surface such as the mean curvature and Gaussian curvatures and we plot the ruled surface with its base curve, and at symmetric points and along a symmetry axis.

In this paper, we introduce ruled like surfaces in three-dimensional Euclidean space, E³ . To form a ruled like surface in E³, we consider a base curve γ(s) and a director curve X(s). Let parameter s be the angle between the tangent of γ(s) and X(s) when X(s) lie on rectifying plane or in the osculating plane. Whereas, if X(s) is in the normal plane, then parameter s will be the angle between the normal of γ(s) and position vector of X(s) at the corresponding point in E³ . Then we investigate some characterizations of such types of surfaces (say S(s, v)). Moreover, we find the condition for the existence of Bertrand mate of γ(s) in S(s, v). Finally, as examples, we construct the surfaces S(s, v) by using a straight line, circle and helix in E³ . [ABSTRACT FROM AUTHOR]

In this paper, some geometric properties of equiform Smarandache ruled surfaces in Minkowski space E1 3 using an equiform frame are investigated. Also, we give the sufficient conditions that make these surfaces are equiform developable and equiform minimal related to the equiform curvatures and when the equiform base curve contained in a plane or general helix. Finally, we provide an example, such as these surfaces. [ABSTRACT FROM AUTHOR]

In this paper, we study the singularities on a non-developable ruled surface according to Blaschke's frame in the Euclidean 3-space. Additionally, we prove that singular points occur on this kind of ruled surface and use the singularity theory technique to examine these singularities. Finally, we construct an example to confirm and demonstrate our primary finding. [ABSTRACT FROM AUTHOR]

The study of a family of equiform Bishop spherical image ruled surfaces created by some specific curves such as spherical image in Minkowski 3-space using equiform Bishop frame of that curve is presented in this paper. We also offer the necessary criteria for these surfaces to be equiform Bishop developable and equiform Bishop minimum in relation to equiform Bishop curvatures, as well as when the curve is enclosed in a plane. Finally, we provide an example, such as these surfaces.

Modelling of the CUBE's TWIST shell This paper gives an overview of the main procedures prepared during the phase of the project development within the field of numerical modelling and design. The roof shell elements represent itself twisted shape that together with complicated structurally demanded inner structure forms quite complicated configuration for the concrete volumes as well as for the used reinforcement. Therefore, custom developed numerical method was required to be prepared. For that, the software solutions Rhinoceros 3D and Grasshopper 3D were used as the main algorithmic environment. The paper additionally gives an insight of important aspects required for production of the formwork components, thermal insulation, biaxially oriented carbon reinforcing grids and other parts of the load‐bearing shells. Finally, the paper gives an idea how was established a collaboration with partners of the CUBE project in particular in the field of structural analysis. [ABSTRACT FROM AUTHOR]

There is a unique curve called striction curve that is not present on a surface except a ruled surface. This curve is defined as the shortest distance with the help of a common perpendicular line between two adjacent rulings. In this work, we present Lorentz forces and magnetic striction curves produced by the geodesic Frenet frame ē,t¯,g¯$$ \left\{\bar{e},\overline{t},\overline{g}\right\} $$ on the striction curve of the ruled surface defined by the spherical indicatrix curve in a magnetic field. We calculate magnetic vector fields of magnetic striction curves for e‾,t¯$$ \overline{e},\overline{t} $$, and g¯$$ \overline{g} $$. Furthermore, we define magnetic flux surfaces constructed by magnetic vector fields along magnetic striction curves. We obtain developability conditions for these surfaces according to their curvature functions. Finally, we give some examples about magnetic flux surfaces. [ABSTRACT FROM AUTHOR]

In this study, we give the structure of ruled surfaces parameterized on a product time scale. Mean and Gaussian curvatures of these surfaces are constructed on distinct time scales. Then, we obtain principal curvatures for ruled surfaces. Finally, we will clarify the issue with some examples. [ABSTRACT FROM AUTHOR]

In this article, firstly, the intersection of two ruled surfaces corresponding to two different curves on S²1 is investigated. The conditions for the intersection of these ruled surfaces in R³1 are expressed by theorems with bivariate functions. Then, the intersection of two ruled surfaces corresponding to two different curves on H² is examined. Similarly, the conditions for the intersection of these ruled surfaces in R³1 are shown by some illustrative theorems with bivariate functions. Finally, some examples are given to support the main results. [ABSTRACT FROM AUTHOR]

In this study, firstly, each natural lift curve of the main curve is corresponded to the ruled surface by exploiting E. Study mapping and the relation among the subset of the tangent bundle of unit 2-sphere, TM and ruled surfaces in ℝ³. Secondly, the intersection of two ruled surfaces, which are obtained by using the relation given above, is examined for the condition of the zero-set of λ (u, v) = 0. Then, all redundant and non-redundant solutions of the zero-set are investigated. Furthermore, the degenerate situations (u, v) = 0, where the whole plane is degenerated by the zero-set, are denoted. Finally, some examples are given to verify the results. [ABSTRACT FROM AUTHOR]

This study begins with the construction of type-Π Smarandache ruled surfaces, whose base curves are Smarandache curves derived by rotation-minimizing Darboux frame vectors of the curve in E3. The direction vectors of these surfaces are unit vectors that convert Smarandache curves. The Gaussian and mean curvatures of the generated ruled surfaces are then separately calculated, and the surfaces are required to be minimal or developable. We report our main conclusions in terms of the angle between normal vectors and the relationship between normal curvature and geodesic curvature. For every surface, examples are provided, and the graphs of these surfaces are produced.

In the 3-dimensional Euclidean space E 3 , a quadric surface is either ruled or of one of the following two kinds z 2 = a s 2 + b t 2 + c , a b c ≠ 0 or z = a 2 s 2 + b 2 t 2 , a > 0 , b > 0 . In the present paper, we investigate these three kinds of surfaces whose Gauss map N satisfies the property Δ I I N = Λ N , where Λ is a square symmetric matrix of order 3, and Δ I I denotes the Laplace operator of the second fundamental form I I of the surface. We prove that spheres with the nonzero symmetric matrix Λ , and helicoids with Λ as the corresponding zero matrix, are the only classes of surfaces satisfying the above given property. [ABSTRACT FROM AUTHOR]

The study of a family of equiform Bishop spherical image ruled surfaces created by some specific curves such as spherical image in Minkowski 3-space using equiform Bishop frame of that curve is presented in this paper. We also offer the necessary criteria for these surfaces to be equiform Bishop developable and equiform Bishop minimum in relation to equiform Bishop curvatures, as well as when the curve is enclosed in a plane. Finally, we provide an example, such as these surfaces. [ABSTRACT FROM AUTHOR]

Let B^ be a Bézier curve in D3, that is, the control points of B^ are the vectors in D3 = {u = (u1, u2, u3) : ui ∈ D,i = 1,2,3} which is the set of dual vectors whose each coordinate components are dual numbers defined as a+εa* : a,a* ∈ R,ε ≠ 0; ε2 = 0. So the spherical projection of B^ is a spherical curve denoted by B~ on the unit sphere in D3 and every point of B~ corresponds to a directed line in real space R3 by Study transformation. In this study, the ruled surface X(t, ν) corresponding to this projection curve B~t of dual Bézier curve B^t is stated in terms of the parametric equation of the real and dual part of a given dual Bézier curve B^t. Also, in this study, some fundamental characteristics such as the striction curve, the Gaussian curvature, and the distribution parameter of the ruled surface X(t, ν) corresponding to this projection curve B~t of the dual Bézier curve B^t are investigated. These concepts at any point are stated in terms of the control points of the given dual Bézier curve B^t. [ABSTRACT FROM AUTHOR]

In this paper, we classify locally lightlike pseudo-isotropic Lorentzian Calabi surfaces in R 3 . We give expressions of those surfaces and find that some of them are part of quadrics. Since most of these surfaces are ruled surfaces, we also study conditions for a ruled surface to be lightlike pseudo-isotropic. [ABSTRACT FROM AUTHOR]

In this paper, we realize a rapid, smooth iterative morphing algorithm between two immersed and closed spherical curves of arbitrary turning numbers. By compensating in turning numbers, we linearly interpolate between geodesic curvatures and the side lengths of their accurate spherical polygons approximation. Our algorithm is essentially based on a geometrical closure condition for a flow of spherical polygons involving the variation of both sides and exterior angles. A curvature smoothing flow using discrete geodesic curvature is adopted for smooth curve generation. To demonstrate the good properties of the method, a morphing of spherical curves and ruled surfaces is illustrated. [ABSTRACT FROM AUTHOR]

As a continuation to our results in [1], we study the dual ruled surface defined on the set of dual numbers. The idea of the dual part are defined similar to quaternion space. The dual part of this represents a ruled dual submanifold. The geometric properties of their dual parts are investigated. Some examples are given and plotted.

In this paper, parallel q-equidistant ruled surfaces are defined such that the binormal vectors of given two differentiable curves are parallel along the striction curves of their corresponding binormal ruled surfaces, and the distance between the asymptotic planes is constant at proper points, which is related to symmetry. The characterizations and some other useful relations are drawn for these surfaces as well. If the surfaces are considered to be closed, then the integral invariants such as the pitch, the angle of the pitch, and the drall of them are given. Finally, some examples are presented to indicate that the distance between the proper points on the corresponding asymptotic planes is always constant. [ABSTRACT FROM AUTHOR]

In this paper we define a new family of ruled surfaces using an othonormal Sannia frame defined on a base consisting of the striction curve of the tangent, the principal normal, the binormal and the Darboux ruled surface. We examine characterizations of these surfaces by first and second fundamental forms, and mean and Gaussian curvatures. Based on these characterizations, we provide conditions under which these ruled surfaces are developable and minimal. Finally, we present some examples and pictures of each of the corresponding ruled surfaces. [ABSTRACT FROM AUTHOR]

The parallel curves are especially used in numerically controlled machining where a biaxial machine defines the shape of the cut made with a round cutting tool. In this study, trajectory ruled surfaces are investigated whose base curve is the parallel curve of a prescribed curve. Some results related with their developability and minimality are obtained. In addition, the conditions are investigated for the base curve to be a special curve. Parallel trajectory ruled surfaces are illustrated with examples. [ABSTRACT FROM AUTHOR]

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In this article, we present a procedure to derive algebraic descriptions from geometric descriptions of trimmed hyperbolic paraboloid (or 'hypar') surfaces. We contextualise this procedure historically, and we illustrate its application using the 1958 Philips Pavilion by Le Corbusier and Iannis Xenakis as a case study. The procedure uses parametric modelling and computational optimisation to converge on close algebraic approximations of hyperbolic paraboloid geometry through a successive breakdown of vast search spaces. It departs from coordinate data of three or four vertices of a geometrically described hyperbolic paraboloid and yields the surface's two quadratic coefficients, the coordinates of its centroid location and the rotation angles of its spatial orientation. The procedure exemplifies the under-explored analytical (as opposed to generative) use of computational optimisation and parametric modelling in the field of architectural computing. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the characterization for darboux curves in a strictWalker 3-manifold and a classification for ruled surface, which is generated by darboux curve in Walker 3-manifold is obtained. An example is given at the end. [ABSTRACT FROM AUTHOR]

Some situations that change the parameters of the kinematic structure may cause the robot end effector to deviate [Merlet [2005] Parallel Robots, Vol. 128 (Springer Science & Business Media, Germany)] from the desired trajectory. This effect is called the robustness of the robot by Merlet. One of the ways to correct the robustness is by updating the robot trajectory. The jerk vector of the robot end effector is the third-order positional variation of the TCP and defined as thus the time derivative of the acceleration vector. If there is a high curvature on the transition curve trajectory of robot, then there is a tangential jerk along the trajectory. In this study, the geometrically offset trajectory of the robot end effector from the current trajectory was obtained by using the curvature theory. The angular velocity and angular acceleration of the offset trajectory were calculated. An example of the main trajectory of robot end effector and its offset is given. Also, the jerk of the robot end effector of the offset trajectory was calculated according to the curvature of the trajectory surface in case of a jerk problem caused by a high curvature in the transition curve along the offset trajectory curve. [ABSTRACT FROM AUTHOR]

In this paper, we establish the necessary and sufficient conditions to parameterize a surface family on which the Bertrand D-partner of any given curve lies as isogeodesic, isoasymptotic or curvature line in E3. Then, we calculate the fundamental forms of these surfaces and determine the developability and minimality conditions with the Gaussian and mean curvatures. We also extend this idea on ruled surfaces and provide the required conditions for those to be developable. Finally, we present some examples and graph the corresponding surfaces. [ABSTRACT FROM AUTHOR]

We show that a smooth embedded paper Moebius band must have aspect ratio at least λ 1 = 2 4 - 2 3 + 4 3 4 2 + 2 2 3 - 3 = 1.69497 ... This bound comes more than 3/4 of the way from the old known bound of π / 2 = 1.5708 ... to the conjectured bound of 3 = 1.732 ... [ABSTRACT FROM AUTHOR]

In this study, we investigate the evolution of space curves and the special ruled surfaces with modified orthogonal frame. Firstly, we recall the relations between the Serret-Frenet frame and the modified orthogonal frame. Then we investigate the evolution of space curves relative to these relations. Moreover, we determine the first and second fundamental forms, the Gaussian and mean curvatures of the tangent, normal, and binormal ruled surfaces according to the modified frame. In these regards, we obtain the characterizations of the minimal and developable ruled surfaces based on modified orthogonal frame.