arXiv research

The paper studies singularities of pedal curves of hyperbolic frontals.

New Finsler metrics derived from pedal curves.

The pedal of a curve in the Euclidean plane is a classical subject which has a singular point at the inflection point of the original curve or the pedal point. The primitive of a curve is a curve given by the inverse construction for making the pedal. In this paper we consider the pedal of a quadratic curve. On of the …

Pedal curves derived from ellipses are invariant in area.

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…

Study circle families' envelopes and related curves.

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.

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.

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

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…

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

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…

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 (\…

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…

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…

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…

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…