In this study, we define a new type of direction curves in the Euclidean 3-space such as osculating-direction curve. We give the characterizations for these curves. Moreover, we obtain the relationships between osculating direction curves and some special curves such as helix, slant helix or rectifying curves.
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48 results for “direction curves”
A directed curve is a possibly singular curve with well-defined tangent lines along the curve. Then the tangent surface to a directed curve is naturally defined as the ruled surface by tangent geodesics to the curve, whenever any affine connection is endowed with the ambient space. In this paper the local diffeomorphis…
Directional Pliability, Whitney Extension, and Lusin Approximation for Curves in Carnot Groupsmath.DG
In Carnot groups, directional pliability allows curve extensions and approximations.
problem Existence of curve extensions and approximations in Carnot groups.
method Directional pliability in subsets of directions guarantees Whitney-type extensions and Lusin approximations.
result Every horizontal curve in the Engel group intersects a curve in a set of positive measure.
Study of curves and surfaces from single-direction projections.
problem Obtaining complete shape information from a single view.
method Theoretical study of differential geometric information from multiple orthogonal projections.
result Formulae for recovering certain information on curves or surfaces from their projections.
New methods to construct curve pairs and their applications.
problem Constructing curve pairs and their properties.
method Using integral curves to study direction and donor curves.
result New methods to construct partner curves of unit speed curves.
New method to construct partner curves of non-lightlike curves.
problem Constructing partner curves for non-lightlike curves.
method Using integral curves in Minkowski 3-space, direction curve, and donor curve.
result New methods to construct partner curves of a unit speed non-lightlike curve.
The paper studies circular evolutes and involutes of framed curves in Euclidean space.
problem Investigating properties of framed curves and their evolutes and involutes.
method Definition and analysis of circular evolutes and involutes of framed curves, properties of normal surfaces, and their relations.
result Circular evolutes and involutes of framed curves are opposite operations under suitable assumptions, similar to fronts in the Euclidean plane.
The Frenet frame is generally known an orthonormal vector frame for curves. But, it does not always meet the needs of curve characterizations. In this study, with the help of associated curves of any spatial curve we obtained a new orthonormal frame which has the property that the second vector makes a constant angle w…
In this paper, we study the spherical indicatrices of W-direction curves in three dimensional Euclidean space which were defined by using the unit Darboux vector field W of a Frenet curve, in [11]. We obtain the Frenet apparatus of these spherical indicatrix curves and the characterizations of being general helix and s…
Permutations encoding the local shape of level curves of real polynomials via generic projectionsmath.AG
The paper encodes local shapes of polynomial curves using permutations.
problem Measuring non-convexity of real algebraic plane curves.
method Generic projections avoiding specific tangencies.
result Local shapes of curves can be encoded in alternating permutations.
A new probabilistic polygonal curve representation using Gaussian Mixture Models.
problem Capturing curves with uncertainty in both tangent and normal directions.
method Probabilistic polygonal approximation with Gaussian Mixture Model (GMM).
result The GMM accurately captures the local geometry and uncertainty of curves.
An analytic approach and description are presented for the moduli cotangent sheaf for suitable stable curve families including noded fibers. For sections of the square of the relative dualizing sheaf, the residue map at a node gives rise to an exact sequence. The residue kernel defines the vanishing residue subsheaf. F…
Paper approximates continuous functions on Jordan arcs using conformal minimal immersions.
problem Approximating continuous functions on Jordan arcs using minimal immersions.
method Conformal minimal immersions and directed holomorphic curves.
result Continuous functions on Jordan arcs can be approximated by conformal minimal immersions.
A theorem connects integral of second-order derivatives to function rise.
problem Understanding the integral of second-order derivatives over regions.
method Proves integral proportional to function rise over specified regions.
result Integral of second-order derivatives equals rise in function value.
Smooth curves from polygonal chains with vertex preservation and explicit curvature control.
problem Preserving vertices while smoothing polygonal chains to curves.
method Directional mollification operator for polygonal chains.
result Smooth curves that intersect original vertices and maintain explicit curvature bounds.
The fine curve graph is hyperbolic and contains all countable graphs as induced subgraphs.
problem Characterizing the structure and properties of fine curve graphs.
method Analyzing the hyperbolicity and induced subgraph properties of fine curve graphs and their direct limits.
result The finitary curve graph has diameter 2, contains every countable graph as an induced subgraph, and has the homeomorphism group of the surface as its automorphism group.
We give the complete solution to the local diffeomorphism classification problem of generic singularities which appear in tangent surfaces, in as wider situations as possible. We interpret tangent geodesics as tangent lines whenever a (semi-)Riemannian metric, or, more generally, an affine connection is given in an amb…
We construct normal forms for Lorentzian metrics on Engel distributions under the assumption that abnormal curves are timelike future directed Hamiltonian geodesics. Then we indicate some cases in which the abnormal timelike future directed curve initiating at the origin is geometrically optimal. We also give certain e…
In this paper, we consider a regular curve on an oriented surface in Euclidean 3-space with the Darboux frame along the curve, where is the unit tangent vector field of the curve, is the surface normal restricted to the curve and $\mathsf{V}=\mathsf{U}\ti…
The study reveals non-homotopy equivalent subspaces of curves with curvature constraints.
problem Understanding the homotopy type of subspaces of curves with curvature constraints.
method Used a version of the h-principle to prove results.
result Explicit construction of exotic generators for some homotopy and cohomology groups.
The paper shows how to rearrange arcs to form closed curves.
problem Creating closed curves from planar arcs.
method Splitting a curve into arcs and rearranging them to form a closed curve.
result Closed curves can be formed by rearranging arcs under weak assumptions.
Study of curves and surfaces in Riemannian spaces making a constant angle with a parallel transported direction.
problem Understanding geometric properties of curves and surfaces in Riemannian spaces.
method Developing a theoretical framework to study curves and surfaces by their angle with a parallel transported vector field.
result Surfaces making a constant angle with a parallel transported direction are extrinsically flat ruled surfaces.
In this paper we consider the problem of reconstructing a curve that is partially hidden or corrupted by minimizing the functional , depending both on length and curvature . We fix starting and ending points as well as initial and final directions. For this functional we discuss the problem o…
Defines new curves from tangent indicatrix of curves, linking them to helices and slant helices.
problem Understanding and constructing helices and slant helices from spherical curves.
method Defining integral curves of Frenet vectors and using their curvatures.
result Established relationships and methods to create helices and slant helices from specific spherical curves.
Let be a complete flat surface, such as the Euclidean plane. We obtain direct characterizations of the connected components of the space of all curves on which start and end at given points in given directions, and whose curvatures are constrained to lie in a given interval, in terms of all parameters involved.…
We consider a pseudo-Riemannian metric that changes signature along a smooth curve on a surface, called the discriminant curve. The discriminant curve separates the surface locally into a Riemannian and a Lorentzian domain. We study the local behaviour and properties of geodesics at a point on the discriminant where th…
This paper defines directional derivatives and solves Maxwell's equations in curved 3D space.
problem Analyzing electromagnetic fields in curved non-flat 3D space.
method Defined directional derivatives and used Frenet formulas to express Serret-Frenet relations. Solved Maxwell's equations for electric and magnetic fields.
result Solved Maxwell's equations for electromagnetic fields in curved 3D space.
The paper is devoted to differential geometric invariants determining a Frenet curve in up to a direct similarity These invariants can be presented by the Euclidean curvatures in terms of an arc lengths of the spherical indicatrices. Then, these invariants expressed by focal curvatures of the curve. And then, we give t…
Direct approach to solving Gauss diagram realizability problem.
problem Which Gauss diagrams can be realized by knots?
method Interpreted conditions as 'number of exits = number of entrances' and based on Jordan Curve Theorem.
result Sufficient conditions for realizability of a Gauss diagram.
Ordering curves on surfacesmath.GT
Study on ordering geodesics on hyperbolic surfaces.
problem Determining the order of lengths of closed geodesics on hyperbolic surfaces.
method Using Teichmüller space and properties of curves on pairs of pants.
result Order of lengths of curves determine a point in Teichmüller space and identify classes of curves with constant order.
We give a simple characterization of the parabolic geodesics introduced by Cap, Slovak and Zadnik for all parabolic geometries. This goes through the definition of a natural connection on the space of Weyl structures. We then show that parabolic geodesics can be characterized as the following data: a curve on the manif…
Characterizes K-polystability for projective bundles over curves.
problem Existence of extremal Kähler metrics on projective bundles.
method Relative K-polystability and decomposition of vector bundles.
result Existence of extremal Kähler metrics linked to K-polystability.
New CRB derived for curved models using extrinsic geometry.
problem Estimate curved statistical families accurately.
method Vector generalization of CRB with curvature correction using SDP and SOS relaxations.
result Directional curvature correction provides more accurate estimation.
Formula for analytic torsion forms in fibrations by projective curves.
problem Calculating analytic torsion forms for specific geometric structures.
method New description of Bismut's equivariant Bott-Chern current for isolated fixed points.
result Explicit formula for analytic torsion forms in fibrations by projective curves.
Sharp chord-arc estimates for curve shortening flow on spheres.
problem Understanding the behavior of curves on spheres under curve shortening flow.
method Proving sharp chord-arc estimates and curvature control.
result Simple spherical curves either contract to points or converge to great circles.
The Mittag-Leffler theorem is extended to meromorphic curves and minimal surfaces.
problem Extending the Mittag-Leffler theorem to meromorphic curves and minimal surfaces.
method Established a Mittag-Leffler-type theorem for meromorphic curves and minimal immersions, including interpolation and approximation.
result Complete minimal ends in R^5 are generically embedded, and open Riemann surfaces are characterized for minimal surfaces.
Unique ancient solutions found for anisotropic curve shortening flow.
problem Finding unique solutions for anisotropic curve shortening flow.
method Constructing translating and ancient solutions under given conditions.
result Unique ancient and translating solutions found for anisotropic curve shortening flow.
The paper analyzes discrete approximations to minimize curve length in Euclidean space.
problem Minimizing the length of curves between two sets in Euclidean space.
method Finite differences and numerical integration for discrete approximations.
result The squared length of the reconstructed curve converges to the squared minimal length with rate .
The paper examines how closed curves on surfaces intersect and how this intersection determines the curves.
problem Determining closed curves on surfaces based on their intersections.
method Constructing and studying -equivalent curves, analyzing intersections with other curves.
result Curves are determined by their intersections with all other curves, but non-simple curves require infinitely many intersections to distinguish.
We study the geometry of curves in the Minkowski space and in the de Sitter space, specially at points where the tangent direction is lightlike (i.e. has length zero) called lightlike points of the curve. We define the focal sets of these curves and study the metric structure of them. At the lightlike points, the focal…
The paper proves interpolation of minimal surfaces and holomorphic curves.
problem Interpolating minimal surfaces and holomorphic curves on Riemann surfaces.
method Using conformal minimal immersions and directed holomorphic curves.
result One can prescribe values of conformal minimal immersions and directed holomorphic curves on closed discrete subsets of Riemann surfaces.
Optimal flat ribbons can be created from nonplanar curves with minimal energy.
problem Finding the optimal shape of flat ribbons from nonplanar curves.
method Direct method of the calculus of variations.
result Optimal flat ribbons can be created with minimal bending energy, but they may have isolated planar points.
The paper classifies helix curves on a pseudo-Riemannian surface.
problem Classifying helix curves on pseudo-Riemannian surfaces.
method Analyzing geodesic flow vector fields and pseudo-Riemannian metrics.
result All helix curves are circular helixes with constant curvature and torsion.
The study explores Bertrand and Mannheim curves in 4D Euclidean space for framed curves.
problem Exploring Bertrand and Mannheim curves in 4D Euclidean space for framed curves.
method Defining and investigating Bertrand and Mannheim curves of framed curves in 4D Euclidean space.
result Bertrand and Mannheim curves exist even for framed curves in 4D Euclidean space, contrary to regular curves.
Study the boundary of hyperbolic groups generated by atoroidal automorphisms.
problem Characterize the Gromov boundary of hyperbolic groups generated by atoroidal automorphisms.
method Define directional Whitehead graphs and prove properties of indecomposable trees. Use these to show boundary homeomorphism to Menger curve.
result The boundary of hyperbolic groups generated by atoroidal, fully irreducible automorphisms is homeomorphic to the Menger curve.
Study inextensible flows of curves in 4D pseudo-Galilean space and defines energy functions.
problem Analyzing inextensible flows and energy of curves in 4D pseudo-Galilean space.
method Expressed inextensible flows as partial differential equations, defined directional derivatives, and expressed bending elastic energy functions.
result Necessary and sufficient conditions for inextensible flows are given as partial differential equations.
Study slopes of direct images in complex manifolds, proving a Mehta-Ramanathan type theorem.
problem Distribution of Harder-Narasimhan slopes in direct image sheaves.
method Analyzing asymptotic distributions of slopes under base changes of families of complex projective manifolds.
result Asymptotic distribution of slopes can be recovered from base changes over generic curves.
Study shows Lelong numbers vanish for certain currents in weakly hyperbolic foliations.
problem Analyzing Lelong numbers for currents in weakly hyperbolic foliations.
method Local and global analysis of directed positive harmonic currents and currents directed by foliations.
result Lelong numbers of currents at the singularity vanish.