Circular evolutes and involutes of framed curves in the Euclidean space

A characterization of involutes and evolutes of a given curve in $\mathbb{E}^{n}$

The orthogonal trajectories of the first tangents of the curve are called the involutes of $x$. The hyperspheres which have higher order contact with a curve $x$ are known osculating hyperspheres of $x$. The centers of osculating hyperspheres form a curve which is called generalized evolute of the given curve $x$ in $n$-dimensional Euclidean space $\mathbb{E}^{n}$. In the present study, we give a characterization of involute curves of order $k$ (resp. evolute curves) of the given curve $x$ in $n$-dimensional Euclidean space $\mathbb{E}^{n}$. Further, we obtain some results on these type of cur…

Some characterizations for the involute curves in dual space

In this paper, we investigate some characterizations of involute -- evolute curves in dual space. Then the relationships between dual frenet frame and darboux vectors of these curves are found.

Iteration of Involutes of Constant Width Curves in the Minkowski Plane

In this paper we study properties of the area evolute (AE) and the center symmetry set (CSS) of a convex planar curve $γ$. The main tool is to define a Minkowski plane where $γ$ becomes a constant width curve. In this Minkowski plane, the CSS is the evolute of $γ$ and the AE is an involute of the CSS. We prove that the AE is contained in the region bounded by the CSS and has smaller signed area. The iteration of involutes generate a pair of sequences of constant width curves with respect to the Minkowski metric and its dual, respectively. We prove that these sequences are converging to symmetr…

Iterating evolutes and involutes

We study iterations of two classical constructions, the evolutes and involutes of plane curves, and we describe the limiting behavior of both constructions on a class of smooth curves with singularities given by their support functions. Next we study two kinds of discretizations of these constructions: the curves are replaced by polygons, and the evolutes are formed by the circumcenters of the triples of consecutive vertices, or by the incenters of the triples of consecutive sides. The space of polygons is a vector bundle over the space of the side directions, and both kinds of evolutes define…

Spherical Indicatrices of Involute of a Space Curve in Euclidean 3-Space

In this work, we studied the properties of the spherical indicatrices of involute curve of a space curve and presented some characteristic properties in the cases that involute curve and evolute curve are slant helices and helices, spherical indicatrices are slant helices and helices and we introduced new representations of spherical indicatrices.

欧几里得三维空间中的Bertrand标架曲面及其应用

Bertrand framed surfaces in the Euclidean 3-space and its applications

本文利用活动标架将Bertrand曲线概念推广到曲面，定义Bertrand标架曲面并研究其焦散面与渐伸面，给出二者互为逆运算的条件，还提出切方向标架曲面这一新运算。

On the natural lift curves for the Involute spherical indicatrices in Minkowski 3-space

The aim of this paper is to determine criteria of being integral curve for the geodesic spray of the natural lift curves of the spherical indicatrices of the involutes of a given spacelike curve with a timelike binormal in Minkowski 3-space. Furthermore, some interesting results about the spacelike evolute curve with timelike binormal and spacelike or timelike Darboux vector were obtained, depending on the assumption that the natural lift curves of the spherical indicatrices of the involute curve should be the integral curve on the tangent bundle. Additionally we illustrate an example of our m…

三维反德西特空间中伪球面标架浸入的焦集奇点

Singularities of focal sets of pseudo-spherical framed immersions in the three-dimensional anti-de Sitter space

本文在三维反德西特时空中引入伪球面非零标架曲线，利用活动标架定义渐屈线和焦曲面，揭示渐屈线恰为焦曲面的奇点集，并通过勒让德奇点理论将其解释为波前。

从Legendrian对偶视角看双曲三维空间中带标架曲线的焦面与渐屈线

Focal surfaces and evolutes of framed curves in hyperbolic 3-space from the viewpoint of Legendrian duality

该研究针对双曲三维空间中可能带奇点的带标架曲线，借助Chen和Izumiya提出的Legendrian对偶性研究其焦面与渐屈线，给出相关对偶曲面的奇点分类，明确三类曲面的关联及奇点对偶关系。

非抛物空间混合框架曲线及其在空间混合数空间中的应用

Non-parabolic Spatial Hybrid Framed Curves and Their Applications in the Spatial Hybrid Number Space

本文定义了可能具有奇点的空间混合框架曲线，并证明了其存在性和唯一性定理，同时探讨了其演化曲线、involute、 pedal 和 contrapedal 曲线的关系。

Notes on Degenerate Curves in Pseudo-Euclidean Spaces of Index Two

In this paper we deal with curves with degeneration degree two in pseudo-Euclidean spaces of index two. We characterize Bertrand curves. We show a correspondence between the evolute of a null curve and the involute of a certain spacelike curve in the $6-$dimensional pseudo-Euclidean space of index two. Also we characterize pseudo-spherical null curves in the $n-$dimensional pseudo-Euclidean space of index two in terms of the curvature functions.

Bertrand光锥标架曲线在Lorentz-Minkowski三维空间中的研究

Bertrand lightcone framed curves in the Lorentz-Minkowski 3-space

本文研究Lorentz-Minkowski三维空间中混合类型曲线，引入光锥标架定义Bertrand光锥标架曲线，阐明其存在条件，并发现混合类型曲线的伪圆渐伸线和渐屈线。

空间曲线的微分几何：被遗忘的篇章

Differential geometry of space curves: Forgotten chapters

本文以现代统一视角重新梳理空间曲线渐屈线与渐伸线的经典理论，补充新观察并利用计算机图形进行可视化说明。

伯特兰勒内曲线在欧几里得平面单位切束上的研究

Bertrand Legendre curves in the unit tangent bundle over Euclidean plane

本文研究了正则平面曲线及其勒内曲线的关联曲线，探讨了伯特兰正则曲线和伯特兰勒内曲线的性质，提出存在条件及逆操作，并定义了勒内曲线集之间的映射，证明其在等价关系下的双射性。

广义Frenet标架与奇异空间曲线的标架序列

Generalized Frenet frames and frame sequences of singular space curves

针对经典Frenet标架在奇异点和曲率零点失效的问题，基于球面前沿理论和Legendre对偶，引入奇异空间曲线的广义Frenet标架及标架序列，统一了Frenet、Bishop标架和球面前沿的渐屈线-渐伸线对应，并导出递推公式。

欧几里得空间中作为旋转曲面变形的前曲线螺旋面及其奇异性

Helicoidal surfaces of frontals in Euclidian space as deformations of surfaces of revolution, with singularities

本文利用 Legendre 曲线与标架曲面理论研究带奇异性的前曲线螺旋面，给出其成为 frontal/front 的条件及曲率公式，并证明平行曲面与焦曲面的螺旋性及奇点稳定性。