We investigate the relationship among characteristic curves on developable surfaces. In case parameter curves coincide with these curves, we show that the base curve of a developable surface could be either a plane curve, a circular helix, a general helix or a slant helix.
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48 results for “development of curves”
The paper studies singularities on parallels of tangent developable surfaces of frontal curves.
problem Understanding singularities on parallels of tangent developable surfaces.
method Generalization of tangent developable surfaces and parallel deformations for frontal curves in arbitrary dimensions.
result Classification of generic singularities on parallels of tangent developable surfaces for frontal curves in 3 or 4 dimensional Euclidean spaces.
A Condition Ensuring Spatial Curves Develop Type-II Singularities Under Curve Shortening Flowmath.DG
We provide a condition for spatial curves which rules out the development of a type I singularity. The condition is that after the last time for which an inflection point develops, if the torsion is ever everywhere non-negative, the curve cannot develop a type I singularity.
Curves can bound only finitely many developable surfaces.
problem Bounding developable surfaces with nonvanishing mean curvature.
method Proof of finiteness for developable surfaces with prescribed boundary curves.
result Generic curves bound only finitely many developable surfaces.
Paper develops invariants for spherical curves using chord diagrams.
problem Developing invariants for spherical curves under local moves.
method Using based chord diagrams and local moves from Reidemeister moves.
result Invariants include both classical and new spherical curve invariants.
We develop the structure theory of full isometry groups of locally compact non-positively curved metric spaces. Amongst the discussed themes are de Rham decompositions, normal subgroup structure and characterising properties of symmetric spaces and Bruhat--Tits buildings. Applications to discrete groups and further dev…
In this study, we have obtained the distribution parameter of a ruled surface generated by a straight line in Frenet trihedron moving along a timelike curve and also along another curve with the same parameter. At this time, the Frenet frames of these timelike curves are not the same. We have moved the director vector …
Paper proves existence of curves with specific geometric properties.
problem Existence of isometric immersions with prescribed second fundamental form.
method Introducing developments of curves with symmetric tensors and geometric construction.
result Existence of isometric immersions with prescribed second fundamental form.
Study on curve shortening flow in 3D space curves, showing convexity preservation and avoidance principle.
problem Analyzing the behavior of space curves under curve shortening flow in .
method Analysis of properties of space curves evolved by the curve shortening flow, including convexity preservation and avoidance principle.
result Orthogonal projections of space curves remain convex, and the Avoidance principle is shown for spherical curves.
Developable ruled surfaces generated by curvature axes of curves.
problem Creating simple and understandable ruled surfaces for practical design.
method Investigating a straightforward method to generate developable ruled surfaces using curvature axes of curves.
result Developable ruled surfaces are generated by the curvature axes of curves, and these surfaces are developable.
Study on curves around a Whitney umbrella focusing on geodesic and normal curvatures.
problem Analyzing geometric properties of curves around a specific surface.
method Examined geodesic and normal curvatures, ruled surfaces, and normal developable surfaces.
result Obtained functions representing geometry on a Whitney umbrella.
Study curves on a Whitney umbrella using geometric invariants.
problem Analyzing curves passing through a specific geometric shape.
method Using Darboux frame and Frenet-Serre formulas, define invariants and investigate their properties.
result Identified degrees of divergence and top-terms of invariants.
Develops intrinsic curved cosets for Cartan geometries.
problem Defines curved cosets for arbitrary Cartan geometries.
method Defines intrinsic holonomy group and curved cosets.
result Curved cosets retain characteristics of homogeneous counterparts and behave well under automorphisms.
We investigate the evolution of open curves with fixed endpoints under the curve shortening flow, which evolves curves in proportion to their curvature. Using a distance comparison of Huisken, we determine the long-term behavior of open curves with fixed endpoints evolving in certain convex domains on surfaces of const…
Extended de Rham theorem for manifolds with boundaries.
problem Applying de Rham decomposition to manifolds with boundaries.
method Using development of curves to extend classical theorem.
result Extended de Rham theorem for manifolds with boundaries.
Two flows for convex curves converge to circles smoothly.
problem Curvature flow of convex closed plane curves.
method 1/κ^n type curvature flows for closed convex planar curves.
result Closed convex curves converge to a circle smoothly.
We develop and apply an approach for analyzing multi-curve data where each curve is driven by a latent state process. The state at any particular point determines a smooth function, forcing the individual curve to switch from one function to another. Thus each curve follows what we call a switching nonparametric regres…
Spherical quadrilaterals classified based on geometric properties.
problem Classifying spherical quadrilaterals up to isometry.
method Developing map and condition of genericity.
result Space of quadrilaterals with prescribed angles consists of finitely many open curves.
Lie algebroids and curved Lie algebras are equivalent categories.
problem Understanding the relationship between Lie algebroids and curved Lie algebras.
method Developed a method to study the -category of curved Lie algebras using homotopy theory of algebras over a complete operad.
result Equivalence of -categories between Lie algebroids and certain kinds of curved Lie algebras.
We develop an essentially algebraic method to study biharmonic curves into an implicit surface. Although our method is rather general, it is especially suitable to study curves into surfaces defined by a polynomial equation: in particular, we use it to give a complete classification of biharmonic curves into real quadr…
New formulas for Bertrand curves lead to harmonicity conditions.
problem Understanding harmonicity of Bertrand curves.
method Developed new Frenet formulas and used them to write differential equations and harmonicity conditions.
result Sufficient conditions for harmonicity of Bertrand curves expressed in terms of the main curve.
Paper explores folding patterns of curved creases preserving their geometric properties.
problem Investigating rigid-ruling folding motions of curved crease-rule patterns.
method Deriving conditions for rigid-ruling foldability and analyzing combinations of creases.
result Constant fold-angle creases are only compatible with other constant fold-angle creases.
Study geometry of surfaces glued along a curve.
problem Geometry of surfaces glued along a curve.
method Moving frame along the curve, developable surfaces defined.
result Developed geometric properties of glued surfaces.
The field of multiple view geometry has seen tremendous progress in reconstruction and calibration due to methods for extracting reliable point features and key developments in projective geometry. Point features, however, are not available in certain applications and result in unstructured point cloud reconstructions.…
The paper proves the existence of a folded annulus with multiple creases.
problem Existence of a folded annulus with multiple creases.
method Analyze developable surfaces, use normal curvature and relative torsion, compute geometric descriptors, prove propagation of folds.
result Proves the existence of a folded annulus with multiple creases.
We construct explicit solutions to the discrete motion of discrete plane curves that has been introduced by one of the authors recently. Explicit formulas in terms the function are presented. Transformation theory of the motions of both smooth and discrete curves is developed simultaneously.
Finite-Gap Solutions of the Pohlmeyer--Lund--Regge Equation and the Associated Curve Evolutionmath.DG
Develops finite-gap solutions for Pohlmeyer--Lund--Regge equation and Lund--Regge curve evolution.
problem Solving the Pohlmeyer--Lund--Regge equation and understanding Lund--Regge curve evolution.
method Finite-gap construction using hyperelliptic spectral data, Baker--Akhiezer function, and -frame.
result Explicit theta-quotient formula for PLR solutions and criteria for Lund--Regge curve evolution.
Surfaces and curves play an important role in geometric design. In recent years, problem of finding a surface passing through a given curve has attracted much interest. In the present paper, we propose a new method to construct a surface interpolating a given curve as the asymptotic curve of it. Also, we analyze the co…
Unified description of aesthetic curves through self-affinities.
problem Characterizing log-aesthetic curves and their properties.
method Reformulating and proving self-affinities of planar curves, integrating equiaffine geometry.
result Unified characterization of constant curvature curves in similarity and equiaffine geometries.
New conditions found for hyperbolic bicycle tracks.
problem Conditions for hyperbolic bicycle tracks.
method Hyperbolic development interpretation of bicycling monodromy.
result New necessary and sufficient conditions for hyperbolic bicycle tracks.
Shape analysis methods have in the past few years become very popular, both for theoretical exploration as well as from an application point of view. Originally developed for planar curves, these methods have been expanded to higher dimensional curves, surfaces, activities, character motions and many other objects. In …
This study defines finite-type invariants for curves on surfaces and reveals the construction of these finite-type invariants for stable homeomorphism classes of curves on compact oriented surfaces without boundaries. These invariants are a higher-order generalisation of a part of Arnold's invariants that are first-ord…
Study of curve shortening flow for twisted curves, defining curvature-torsion entropy.
problem Understanding the behavior of curves with curvature and torsion under curve shortening flow.
method Defined curvature-torsion entropy to analyze the flow of twisted curves.
result Curved curves under curve shortening flow either develop inflection points or exhibit highly irregular singularities.
Develops a new theory of width for embedded circles in Riemannian manifolds.
problem Defining and understanding the width of embedded circles in Riemannian manifolds.
method Morse-Lusternik-Schnirelmann theory applied to geodesics and minimising configurations.
result Classifies configurations of minimising geodesics intersecting embedded circles.
The paper extends curve deformation methods in Minkowski plane.
problem Studying deformations of curves in the Minkowski plane considering their geometry and singularities.
method Extends methods from [17, 18] to analyze 2-parameter families of curves in Minkowski plane.
result Obtains geometry of deformed curves, including inflections, vertices, and lightlike points.
Nelson and Siegel curves are widely used to fit the observed term structure of interest rates in a particular date. By the other hand, several interest rate models have been developed such their initial forward rate curve can be adjusted to any observed data, as the Ho-Lee and the Hull and White one factor models. In t…
This is a survey on known results and open problems about Smooth and PL-Rigidity Problem for negatively curved locally symmetric spaces. We also review some developments about studying the basic topological properties of the space of negatively curved Riemannian metrics and the Teichmuller space of negatively curved me…
The paper develops theory for holomorphic null curves in SL2(C).
problem Developing theory for holomorphic null curves in SL2(C).
method Establish Runge, Mergelyan, Mittag-Leffler, and Carleman type theorems for holomorphic null immersions.
result Proves every open Riemann surface admits a proper holomorphic null embedding into SL2(C).
In this paper, we use the distance comparison principle, first been developed by G. Huisken, to study the spatial curve shortening flow. We have got the result that if the initial curve is the helix, then the local minimum of the ratio of the extrinsic and intrinsic distance is non-decreasing. And we have proved a Gray…
New distance comparison principle for curve shortening flow in higher dimensions.
problem Understanding curve shortening flow in higher dimensions.
method Established a variant of Huisken's distance comparison principle.
result Symmetric curve shortening flow with one-to-one convex projection develops Type I singularities and becomes asymptotically circular.
Study shows surfaces without certain curves have infinite orbit graph.
problem Characterizing surfaces with specific curve properties.
method Utilized tools from mapping class group geometry.
result Infinite-invariance index 1 surfaces lack good curve graphs.
In this note, we develop a condition on a closed curve on a surface or in a 3-manifold that implies that the curve has the property that its length function on the space of all hyperbolic structures on the surface or 3-manifold completely determines the curve. For an orientable surface of negative Euler characteris…
The theory of classical types of curves in normed planes is not strongly developed. In particular, the knowledge on existing concepts of curvatures of planar curves is widespread and not systematized in the literature. Giving a comprehensive overview on geometric properties of and relations between all introduced curva…
Develops a new model for multiple yield curves using branching processes.
problem Reproduce empirical features of spreads between interbank rates.
method Continuous-state branching processes with immigration (CBI processes).
result Models can generate contagion effects among different spreads.
The delta invariant of curves on rational surfaces II: Poincaré series and topological aspectsmath.GT
Study delta invariant of curves on rational surfaces using topological methods.
problem Calculate delta invariant for curves embedded in rational singularities.
method Use topological techniques and Poincaré series.
result Develop formulae for delta invariant in terms of embedded data.
The main purpose of this paper is to show that ideas of deformation theory can be applied to "infinite dimensional geometry". We develop the deformation theory of Brody curves. Brody curve is a kind of holomorphic map from the complex plane to the projective space. Since the complex plane is not compact, the parameter …
Develops a simple model to understand learning curves for arbitrary power laws.
problem Lack of theoretical understanding of scaling laws in machine learning.
method Analyzes a toy model to determine if learning curves are universal or depend on data distribution.
result Determines that learning curves can exhibit for arbitrary power .
Nonrigidity of flat ribbonsmath.DG
Study of flat ribbons constructed along curves in 3D space.
problem Determine the conditions for a ruled structure to form a flat ribbon.
method Investigate the ruled structure of flat ribbons and calculate energy bounds.
result There exists a well-defined flat ribbon only up to an initial condition.