arXiv research

Study on cubic curves and their tangents, finding Zariski pairs.

A simple closed curve $γ$ in the real projective plane $P^2$ is called anti-convex if for each point $p$ on the curve, there exists a line which is transversal to the curve and meets the curve only at $p$. We shall prove the relation $i(γ)-2δ(γ)=3$ for anti-convex curves, where $i(γ)$ is the number of independent (true…

We show that every smooth closed curve C immersed in Euclidean 3-space satisfies the sharp inequality 2(P+I)+V >5 which relates the numbers P of pairs of parallel tangent lines, I of inflections (or points of vanishing curvature), and V of vertices (or points of vanishing torsion) of C. We also show that 2(P'+I)+V >3, …

We define a computable topological invariant $μ(γ)$ for generic closed planar regular curves $γ$, which gives an effective lower bound for the number of inflection points on a given generic closed planar curve. Using it, we classify the topological types of locally convex curves (i.e. closed planar regular curves witho…

Study on inflection points of plane curve shadows with fixed embedded shapes.

Research on refined algebraic domains respecting differential geometry.

A method is proposed to construct spiral curves by inversion of a spiral arc of parabola. The resulting curve is rational of 4-th order. Proper selection of the parabolic arc and parameters of inversion allows to match a wide range of boundary conditions, namely, tangents and curvatures at the endpoints, including thos…

We give a global description of envelopes of geodesic tangents of regular curves in (not necessarily convex) Riemannian surfaces. We prove that such an envelope is the union of the curve itself, its inflectional geodesics and its tangential caustics (formed by the conjugate points to those of the initial curve along th…

There is an elegant relation found by Fabricius-Bjerre [Math. Scand 40 (1977) 20--24] among the double tangent lines, crossings, inflections points, and cusps of a singular curve in the plane. We give a new generalization to singular curves in RP^2. We note that the quantities in the formula are naturally dual to each …

The study classifies points on ruled surfaces in 4-space based on geometric properties.

Paper proves unique tangent flow at infinity for entropy-limited curve shortening.

The paper classifies deformations of curves with inflections and vertices.

In this note we study the distribution of real inflection points among the ovals of a real non-singular hyperbolic curve of even degree. Using Hilbert's method we show that for any integers $d$ and $r$ such that $4\leq r \leq 2d^2-2d$, there is a non-singular hyperbolic curve of degree $2d$ in $\mathbb R^2$ with exactl…

We describe the structure of the asymptotic lines near an inflection point of a Lagrangean surface, proving that in the generic situation it corresponds to two of the three possible cases when the discriminant curve has a cusp singularity. Besides being stable in general, inflection points are proved to exist on a comp…

We give a criterion when a planar tree-like curve, i.e. a generic immersed plane curve each double point of which cuts it into two disjoint parts, can be send by a diffeomorphism of the plane onto a curve with no inflection points. We also present some upper and lower bounds for the minimal number of inflection points …

The bridge index and superbridge index of a knot are important invariants in knot theory. We define the bridge map of a knot conformation, which is closely related to these two invariants, and interpret it in terms of the tangent indicatrix of the knot conformation. Using the concepts of dual and derivative curves of s…

At a 3/2-cusp of a given plane curve $γ(t)$, both of the Euclidean curvature $κ_g$ and the affine curvature $κ_A$ diverge. In this paper, we show that each of $\sqrt{|s_g|}κ_g$ and $(s_A)^2 κ_A$ (called the Euclidean and affine normalized curvature, respectively) at a 3/2-cusp is a smooth function of the variable $t$, …

Invariants count inflections and vertices in singular plane curves.

In the previous paper (arXiv:0804.0701), the authors gave criteria for A_{k+1}-type singularities on wave fronts. Using them, we show in this paper that there is a duality between singular points and inflection points on wave fronts in the projective space. As an application, we show that the algebraic sum of 2-inflect…

The SIR model and machine learning predict pandemic inflection and end times.

Let $X\subset \mathbb P^N$ be a scroll over a smooth curve $C$ and let $Ł=\mathcal O_{\mathbb P^N}(1)|_X$ denote the hyperplane bundle. The special geometry of $X$ implies that some sheaves related to the principal part bundles of $Ł$ are locally free. The inflectional loci of $X$ can be expressed in terms of these she…

BioFinBERT analyzes sentiment of biotech press releases and financial text around inflection points.

Forecasting COVID-19 cases in Senegal using machine learning.

Given a plane curve $γ: S^1\to \mathbb R^2$, we consider the problem of determining the minimal number $I(γ)$ of inflections which curves $\mbox{diff}(γ)$ may have, where $\mbox{diff}$ runs over the group of diffeomorphisms of $\mathbb R^2$. We show that if $γ$ is an immersed curve with $D(γ)$ double points and no othe…

Abstract: Study of surface transitions and IDE inflections via contact geometry.

Investigate the local geometry of smooth surfaces in 4-space via contact with 2-planes and apparent contours.

The study proves spherical polygon analogs of curve theorems, finding bounds on intersections and inflections.

The paper proves curves with specific properties have at least 4 sign changes in torsion.

We give optimal lower bounds for the number of sextactic points on a simple closed curve in the real projective plane. Sextactic points are after inflection points the simplest projectively invariant singularities on such curves. Our method is axiomatic and can be applied in other situations.

The indicatrix or curvature ellipse and the characteristic curve of a surface in $\mathbf R^4$ are presented, as well as the projective duality connecting them. The characterisation of points in the surfaces as elliptic, parabolic and hyperbolic points, and the inflection points, are also discussed.

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.

We present a new proof of Reidemeister and Singer's Theorem that any two Heegaard splittings of the same 3-manifold have a common stabilization. The proof leads to an upper bound on the minimal genus of a common stabilization in terms of the number of negative slope inflection points and type-two cusps in a Rubinstein-…

Computes the monodromy of cubic surfaces branching over smooth cubic curves.

Osculating spaces of decomposable scrolls (of any genus and not necessarily normal)are studied and their inflectional loci are related to those of their generating curves by using systematically an idea introduced by Piene and Sacchiero in the setting of rational normal scrolls. In this broader setting the extra compon…

Vogt's theorem, concerning boundary angles of a convex arc with monotonic curvature (spiral arc), is taken as a starting point to establish basic properties of spirals. The theorem is expanded by removing requirements of convexity and curvature continuity; the cases of inflection and multiple windings are considered. P…

Sub-Riemannian geometry connects bike paths to mathematical curves.

The paper finds curves minimizing elastic energy pinned at endpoints.

The paper characterizes pedal curves of quadratic curves.

Two new invariants that are closely related to Milnor's curvature-torsion invariant are introduced. The first, the spiral index of a knot, captures the minimum number of maxima among all knot projections that are free of inflection points. This invariant is closely related to both the bridge and braid index of the knot…

Unified method visualizes curvature on curves and surfaces.

We describe a general family of curved-crease folding tessellations consisting of a repeating "lens" motif formed by two convex curved arcs. The third author invented the first such design in 1992, when he made both a sketch of the crease pattern and a vinyl model (pictured below). Curve fitting suggests that this init…

Study of choosing distinct points on cubic curves, proving impossibility.

Analyzes Gerstner's trochoidal waves and their geometric properties.

The paper extends curve deformation methods in Minkowski plane.

Improved Polish language model using subword tokenization.

Study of curve shortening flow for twisted curves, defining curvature-torsion entropy.

This paper introduces tangent display maps to simplify tangent category theory.

Paper constructs infinitely many tangent functors on diffeological spaces.