arXiv research

The paper characterizes pedal curves of quadratic curves.

The paper studies singularities of pedal curves of hyperbolic frontals.

New Finsler metrics derived from pedal curves.

Pedal curves derived from ellipses are invariant in area.

For the n-dimensional spherical pedal curve $ped_{γ,P}$ with respect to an n-dimensional spherical unit speed curve $γ$ and a given point $P \in S^n$, we define the spherical orthotomic curve of $γ$ relative to the point $P$, and classify singularities of spherical orthotomic curves.

Study circle families' envelopes and related curves.

A Steiner deltoid maintains constant area across all boundary points of an ellipse.

Defines non-parabolic curves in spatial hybrid space with applications.

The calculus correspondence has been known to exist between generic pedal evolutions and generic wave front evolutions. In this paper, we first extend the known results on the calculus correspondence to evolutions with multi-parameters, and then give applications of calculus correspondence. Moreover, we discuss the pos…

In this paper, we introduce the notions of map-germs of pedal unfolding type and normalized Legendrian map-germs; and then we show that the fundamental theorem of calculus provides a natural one to one correspondence between Whitney umbrellas of pedal unfolding type and normalized swallowtails.

In this paper, it is shown that for an $n$-dimensional spherical unit speed curve $γ: I\to S^n$, a given point $P \in S^n$ and a point $s_0$ of the open interval $I$, the spherical orthotomic curve-germ $ort_{γ, P}: (I, s_0)\to S^n$ of $γ$ relative to $P$ is $\mathcal{L}$-equivalent to the spherical pedal curve-germ $p…

Given a simple closed plane curve $Γ$ of length $L$ enclosing a compact convex set $K$ of area $F$, Hurwitz found an upper bound for the isoperimetric deficit, namely $L^2-4πF\leq π|F_{e}|$, where $F_{e}$ is the algebraic area enclosed by the evolute of $Γ$. In this note we improve this inequality finding strictly posi…

Legendre curves are smooth plane curves which may have singular points, but still have a well defined smooth normal (and corresponding tangent) vector field. Because of the existence of singular points, the usual curvature concept for regular curves cannot be straightforwardly extended to these curves. However, Fukunag…

We define the notions of $S_t^1\times S_s^1$-valued lightcone Gauss maps, lightcone pedal surface and Lorentzian lightcone height function of Lorentzian surface in semi-Euclidean 4-space and established the relationships between singularities of these objects and geometric invariants of the surface as applications of s…

Superconformal surfaces in Euclidean space are the ones for which the ellipse of curvature at any point is a nondegenerate circle. They can be characterized as the surfaces for which a well-known pointwise inequality relating the intrinsic Gauss curvature with the extrinsic normal and mean curvatures, due to Wintgen (\…

Gaussian processes are the leading class of distributions on random functions, but they suffer from well known issues including difficulty scaling and inflexibility with respect to certain shape constraints (such as nonnegativity). Here we propose Deep Random Splines, a flexible class of random functions obtained by tr…

Data generated by cars is growing at an unprecedented scale. As cars gradually become part of the Internet of Things (IoT) ecosystem, several stakeholders discover the value of in-vehicle network logs containing the measurements of the multitude of sensors deployed within the car. This wealth of data is also expected t…

Assessment of mental workload in real-world conditions is key to ensure the performance of workers executing tasks that demand sustained attention. Previous literature has employed electroencephalography (EEG) to this end despite having observed that EEG correlates of mental workload vary across subjects and physical s…

The study explores Bertrand and Mannheim curves in 4D Euclidean space for framed curves.

The study examines Bertrand Legendre curves in the unit tangent bundle over Euclidean plane.

Method for generating new curves from plane curves on cylinders.

In this study, we introduce a new approach to curve pairs by using integral curves. We consider the direction curve and donor curve to study curve couples such as involute-evolute curves, Mannheim partner curves and Bertrand partner curves. We obtain new methods to construct partner curves of a unit speed curve and giv…

The paper characterizes curves in pseudo-Galilean 4-space.

In this paper, we introduce a new approach to non-lightlike curve pairs by using integral curves in Minkowski 3-space. We consider direction curve and donor curve to study non-lightlike curve couples such as involute-evolute curves, Mannheim partner curves and Bertrand partner curves. We obtain new methods to construct…

The paper explores Bertrand and framed curves in 3D space.

The paper examines how closed curves on surfaces intersect and how this intersection determines the curves.

Approximating complex curves with simple parametric curves is widely used in CAGD, CG, and CNC. This paper presents an algorithm to compute a certified approximation to a given parametric space curve with cubic B-spline curves. By certified, we mean that the approximation can approximate the given curve to any given pr…

Flow deforms locally convex curves to curves of constant k-order width.

Modified curve shortening flow constructs $λ$-Angenent curve.

Study on CR curves in 3-sphere, focusing on critical curves integration and existence.

Study rectifying curves in 3D multiplicative Euclidean space.

In classical curve theory, the geometry of a curve in three dimensions is essentially characterized by their invariants, curvature and torsion. When they are given, the problem of finding a corresponding curve is known as 'solving natural equations'. Explicit solutions are known only for a handful of curve classes, inc…

Unified description of aesthetic curves through self-affinities.

Primitive curves in handlebodies form a connected complex.

Study on Bertrand lightcone framed curves in Lorentz-Minkowski 3-space.

In this study, we introduce a new type of surface curves called D-type curve. This curve is defined by the property that the unit Darboux vector W0 of a space curve r(s) and unit surface normal n along the curve r(s) satisfy the condition <n,W0>=constant. We point out that a D-type curve is a geodesic curve or an asymp…

Study on triharmonic curves in f-Kenmotsu manifolds.

The paper generalizes rectifying and normal curves in Lorentzian n-space.

New findings on hyperbolicity of fine curve graphs and their subgraphs.

Compact curve solution emerges from non-compact curve.

In this paper we study null Bertrand curves in $R_{1}^{4}$ under the assumption the curve has a Cartan frame. We show that if the derivative vectors of the null Cartan curve in $R_{1}^{4}$ is linearly independent, then this curve is not a Bertrand curve. Since then the already known notion of null Bertrand curves in $R…

Study of $p$-biharmonic curves and their properties.

Homotopy types of curve and arc complexes are studied.

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.

We classify curves in the moduli space of curves that are both Shimura- and Teichmueller curves: Except for the moduli space of genus one curves there is only a single such curve. We start with a Hodge-theoretic description of Shimura curves and of Teichmueller curves that reveals similarities and differences of the tw…

In this paper we consider the idea of Bertrand curves for curves lying on surfaces and by considering the Darboux frames of them we define these curves as Bertrand D-curves and give the characterizations for these curves. We also find the relations between the geodesic curvatures, the normal curvatures and the geodesic…

The paper studies a curve flow preserving anisotropic length for convex curves, leading to a homothetic limit.

Rectangular peg problem solved for many curves.