arXiv research

The paper defines evolutes and involutes for framed curves and their properties.

The paper studies circular evolutes and involutes of framed curves in Euclidean space.

ccc-Autoevolutes are closed curves congruent to their evolutes, constructed via symmetry.

The paper studies curve evolution using the PLR equation and its solutions.

Study evolutes of curves with varying smoothness.

Study on focal surfaces and evolutes of framed curves in hyperbolic 3-space using Legendrian duality.

Abstract In this paper, definition of involute-evolute curve couple in Galilean space is given and some well-known theorems for the involute-evolute curves are obtained in 3-dimensional Galilean space.

Study shows continuous evolution of curves in Fréchet distance.

Study the geometry and dynamics of skew evolutes and involutes, related to bicycle kinematics.

In this paper, we take into account the opinion of involute-evolute curves which lie on fully surfaces and by taking into account the Darboux frames of them we illustrate these curves as special involute-evolute partner D-curves in E3. Besides, we find the relations between the normal curvatures, the geodesic curvature…

The evolute of a smooth curve in an m-dimensional Euclidean space is the locus of centers of its osculating spheres, and the evolute of a spatial polygon is the polygon whose consecutive vertices are the centers of the spheres through the consecutive (m+1)-tuples of vertices of the original polygon. We study the iterat…

In this paper, we get the time evolution equations of the curvature and torsion of the evolving spacelike curves in the Minkowski space. Also, we give inextensible evolutions of timelike ruled surfaces that are produced by the timelike normal and spacelike binormal vector fields of spacelike curve and derive the necess…

Space curves with convex projections evolve smoothly until shrinking to a point.

Optical interpretation of Euler's angle problem for caustics of light rays.

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…

Modern treatment of space curve evolutes and involutes.

Continuous curve evolution depends on initial shape on sphere.

Study circle families' envelopes and related curves.

In this study, after introducing algebraic properties of real quaternions some characterizations of quaternionic involute-evolute curves in Q are obtained. And some results and theorems for quaternionic w-curves are given. Lastly, we illustrate some examples and draw their figures with Mathematica Programme.

The paper explores quaternionic curves using differential geometry.

We consider a regular embedded network composed by two curves, one of them closed, in a convex domain $Ω$. The two curves meet only in one point, forming angle of $120$ degrees. The non-closed curve has a fixed end point on $\partialΩ$. We study the evolution by curvature of this network. We show that the maximal exist…

Study on evolutes and focal surfaces of pseudo-spherical framed immersions in anti-de Sitter space.

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

Extending the earlier results for analytic curve segments, in this article we describe the asymptotic behaviour of evolution of a finite segment of a C^n-smooth curve under the geodesic flow on the unit tangent bundle of a finite volume hyperbolic n-manifold. In particular, we show that if the curve satisfies certain n…

The paper classifies different types of cusps on plane curves.

Study geometric mKdV flows for Legendrian curves in a 3-sphere.

We consider evolution equations for curves in the 3-dimensional sphere $S^3$ that are invariant under the group $SU(2,1)$ of pseudoconformal transformations, which preserves the standard contact structure on the sphere. In particular, we investigate how invariant evolutions of Legendrian and transverse curves induce we…

Develops finite-gap solutions for Pohlmeyer--Lund--Regge equation and Lund--Regge curve evolution.

In this paper, the parallel transport frames over non-lightlike curves in Minkowski 3-space are introduced. Evolution equations of these frames with respect to arc length and time are calculated over the space of these curves. Then the equivalence of the non-linear Schrödinger equation and non-linear heat system to the…

In this study, we investigate the locus of the centers of the Meusnier spheres. Just as focal curve is the locus of the centers of the osculating spheres, we investigate the geometrical interpretation on the locus of the centers of the Meusnier spheres. We proved that if the curve is a principal line, the locus of the …

The paper extends curve deformation methods in Minkowski plane.

The local motion of a null curve in Minkowski 3-space induces an evolution equation for its Lorentz invariant curvature. Special motions are constructed whose induced evolution equations are the members of the KdV hierarchy. The null curves which move under the KdV flow without changing shape are proven to be the traje…

Study proves existence and properties of shrinkers in area-preserving curve-shortening flow.

Study curves evolving by gradient flow of elastic energy, proving existence, smoothing, and convergence.

We relate the total curvature and the isoperimetric deficit of a curve $γ$ in a two-dimensional space of constant curvature with the area enclosed by the evolute of $γ$. We provide also a Gauss-Bonnet theorem for a special class of evolutes.

We introduce variational approximations for curve evolutions in two-dimensional Riemannian manifolds that are conformally flat, i.e.\ conformally equivalent to the Euclidean space. Examples include the hyperbolic plane, the hyperbolic disk, the elliptic plane as well as any conformal parameterization of a two-dimension…

Defining Lorentzian Sabban frame of the unit speed time-like curves on de Sitter 2-space $\mathbb{S}^{2}_{1}$ and introducing space-like height function on the unit speed time-like curves on $\mathbb{S}^{2}_{1}$, the invariants of the unit speed time-like curves on $\mathbb{S}^{2}_{1}$ and geometric properties of de Si…

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.

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 r…

Study curves evolving on hypersurfaces with free boundaries, preserving length.

In this study, we introduce a new approach to curve pairs by using integral curves. We consider the direction curve and donor curve to study curve couples such as involute-evolute curves, Mannheim partner curves and Bertrand partner curves. We obtain new methods to construct partner curves of a unit speed curve and giv…

In the present paper, we define the notions of Lorentzian Sabban frames and de Sitter evolutes of the unit speed space-like curves on de Sitter 2-space $\mathbb{S}^{2}_{1}$. In addition, we investigate the invariants and geometric properties of these curves. Afterwards, we show that space-like Bertrand curves and time-…

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…

The paper shows that energy futures yield curves have an affine geometry.

Study of curve evolution in 2D space forms converging to a circle.

In this paper, we introduce a new approach to non-lightlike curve pairs by using integral curves in Minkowski 3-space. We consider direction curve and donor curve to study non-lightlike curve couples such as involute-evolute curves, Mannheim partner curves and Bertrand partner curves. We obtain new methods to construct…

It is shown that the curvature function satisfies a nonlinear evolution equation under the general curve shortening flow and a detailed asymptotic behavior of the closed curves is presented when they contract to a point in finite time.

We show non-collapsing for the evolution of nearly spherical closed convex curves in \mathbb{R}^2 under power curvature flow using two-point-methods.