Iterating skew evolutes and skew involutes: a linear analog of the bicycle kinematics

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.

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…

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。