平曲线上出现的尖点的分类

Cusp singularities of plane envelopes

We consider smooth 1-parameter families of plane curves tangent to a semicubic parabola, when the curvature radius of their curves at the tangency point vanishes at the cusp point. We find the $\A$-normal form of these families, their envelopes and local patterns near the cusp. We obtain a new codimension 2 singularity of envelopes (two transversal semicubic cusps), and we describe its perestroikas under generic small deformations.

Wigner焦散与欧几里得平面中前向曲线的中心对称集的奇点

Singularities of the Wigner Caustic and the Centre Symmetry Set of Frontal Curves in the Euclidean Plane

将Wigner焦散与中心对称集推广至前向曲线，给出尖点判据、降阶关系及尖点计数不等式推广，并分析奇异平行对的共振传递结果。

The behaviour of curvature functions at cusps and inflection points

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 $t$, where $s_g$ (resp. $s_A$) is the Euclidean (resp. affine) arclength parameter of the curve corresponding to the 3/2-cusp $s_g=0$ (resp. $s_A=0$). Moreover, we give a characterization of the behaviour of the curvature functions $κ_g$ and $κ_A$ at 3/2-cusps. On the other hand, infl…

Co--calibrated $G_2$ structure from cuspidal cubics

We establish a twistor correspondence between a cuspidal cubic curve in a complex projective plane, and a co-calibrated homogeneous $G_2$ structure on the seven--dimensional parameter space of such cubics. Imposing the Riemannian reality conditions leads to an explicit co-calibrated $G_2$ structure on $SU(2, 1)/U(1)$. This is an example of an SO(3) structure in seven dimensions. Cuspidal cubics and their higher degree analogues with constant projective curvature are characterised as integral curves of 7th order ODEs. Projective orbits of such curves are shown to be analytic continuations of Al…