Curves In Euclidean Space Research Articles

A new approach to curve couples with Bishop frame

This paper presents a detailed study of a new generation of the Bishop frame with components including three orthogonal unit vectors, which are tangent vector, normal vector and binormal vector. It is a frame field described on a curve in Euclidean space, which is an alternative to the Frenet frame. It is useful for curves for which the second derivative is not available. Moreover, the conditions which the Bishop frame of one curve coincides with the Bishop frame of another curve are defined. It would be valuable to replicate similar approaches in the Bishop frame of one curve coincides with the Bishop frame of another curve.

ASSOCIATED CURVES ACCORDING TO BISHOP FRAME IN 4-DIMENSIONAL EUCLIDEAN SPACE

Many studies have been doneaccording to different frames of the theory of curves in Euclidean Space. Many scientists have studied frames such as the Frenet frame, Bishop frame, and Adapted frame in this theory. These frames help us in the characterization of curves.In this study, associated curves with the Frenet curve according to the Bishop frame in 4-dimensional Euclidean space are investigated. Direction and rectifying curves of the Frenet curve according to this frame are given.

Construction of G2 continuous curve on point-cloud surface

PurposeThe curve construction on surfaces is becoming more and more important in computer-aided design (CAD), computer graphics (CG) and the other related fields. This problem is often encountered in NC machining, tool path generation, automated fiber placement and so on. However, designing curves on curved surfaces is quite different from constructing a curve in Euclidean space. Therefore, the traditional methods of curve design are not suitable for constructing a continuous curve on surface. The authors need to perform interpolation directly on surface so that the final target curve is embedded into the given surface and also meets the continuous conditions.Design/methodology/approachFirstly, adopting a series of Hermite blending functions, the authors design a space curve passing the given knots on the point-cloud surface. Then, the authors construct a class of directrixes that are adopted to determine vector fields for projection. Finally, a complete G2 continuous curve embedded in point-cloud surfaces is constructed by solving the first-order ordinary differential equations (ODEs).Findings The authors’ main contribution is to overcome the problem of constructing G1 and G2 continuous curves on point-cloud surfaces and the authors’ schemes are based on the projection moving least square (MLS) surfaces and traditional differential geometric.Originality/valueBased on the framework of projection MLS surfaces, a novel method to overcome the problem of constructing G2 continuous curves on point-cloud surfaces is proposed.

On the Normal Stability of Triharmonic Hypersurfaces in Space Forms

This article is concerned with the stability of triharmonic maps and in particular triharmonic hypersurfaces. After deriving a number of general statements on the stability of triharmonic maps we focus on the stability of triharmonic hypersurfaces in space forms, where we pay special attention to their normal stability. We show that triharmonic hypersurfaces of constant mean curvature in Euclidean space are weakly stable with respect to normal variations while triharmonic hypersurfaces of constant mean curvature in hyperbolic space are stable with respect to normal variations. For the case of a spherical target we show that the normal index of the small proper triharmonic hypersphere phi :mathbb {S}^m(1/sqrt{3})hookrightarrow mathbb {S}^{m+1} is equal to one and make some comments on the normal stability of the proper triharmonic Clifford torus.

Li–Yau type inequality for curves in any codimension

For immersed curves in Euclidean space of any codimension we establish a Li–Yau type inequality that gives a lower bound of the (normalized) bending energy in terms of multiplicity. The obtained inequality is optimal for any codimension and any multiplicity except for the case of planar closed curves with odd multiplicity; in this remaining case we discover a hidden algebraic obstruction and indeed prove an exhaustive non-optimality result. The proof is mainly variational and involves Langer–Singer’s classification of elasticae and André’s algebraic-independence theorem for certain hypergeometric functions. We also discuss applications to elastic flows, networks, and knots.

B-Lift curves and its ruled surfaces

In this paper, we have described the B-Lift curve in Euclidean space as a curve obtained by combining the endpoints of the binormal vector of a unit speed curve. Subsequently, we have explored the Frenet frames of the B-Lift curves. Moreover, we have introduced the tangent, normal and binormal surfaces of the B-Lift curve and examined the geometric invariants of these surfaces. Finally, we have investigated the singularities of these surface and visualized the surfaces with MATLAB program.

$ C^* $-partner curves with modified adapted frame and their applications

<abstract><p>In this study, the curve theory, which occupies a very important and wide place in differential geometry, has been studied. One of the most important known methods used to analyze a curve in differential geometry is the Frenet frame, which is a moving frame that provides a coordinate system at each point of the curve. However, the Frenet frame of any curve cannot be constructed at some points. In such cases, it is useful to define an alternative frame. In this study, instead of the Frenet frame that characterizes a regular curve in Euclidean space $ E^3 $, we have defined a different and new frame on the curve. Since this new frame is defined with the aid of the Darboux vector, it is very compatible compared to many alternative frames in application. Therefore, we have named this new frame the "modified adapted frame" denoted by $ \{N^*, C^*, W^*\} $. Then, we have given some characterizations of this new frame. In addition to that, we have defined $ N^* $-slant helices and $ C^* $-slant helices according to $ \{N^*, C^*, W^*\} $. Moreover, we have studied $ C^* $-partner curves using this modified adapted frame. Consequently, by investigating applications, we have established the relationship between $ C^* $-partner curves and helices, slant helices.</p></abstract>

On Tzitzeica Curves According to Q-Frame in Euclidean 3-Space

Tzitzeica curve is a special spatial curve, which introduced Gheorghe Tzitzeica in 1911. Gheorghe Tzitzeica defined this curve as follows; for a spatial curve ϕ, "if the ratio of its torsion τ and the square of the distance d from the origin to the osculating plane at an arbitrary point of the curve is constant, then this spatial curve is a Tzitzeica curve in Euclidean space." Moreover Gheorghe Tzitzeica defined a special surface which is named Tzitzeica surface in 1907. In this surface, the asymptotic lines of Tzitzeica surfaces with negative Gaussian curvature are Tzitzeica curves. Also, the ratio of its Gaussian curvature K and the distance d from the origin to the tangent plane at any arbitrary point of the surface is constant. In this paper, we study the Tzitzeica curve via q-frame in E^3. Firstly, we redefine the Tzitzeica curve q-frame in Euclidean 3- space. Then, it is obtained some conditions for the Tzitzeica curve as a spherical curve. Finally, we investigate harmonic vector cases of the Tzitzeica curve according to q- frame in Euclidean space.

A New Approach For Smarandache Curves

In this paper, we introduce new adjoint curves which are associated curves in Euclidean space of three dimension. They are generated with the help of integral curves of special Smarandache curves. We attain some connections between Frenet apparatus of these new adjoint curves and main curve. We characterize these curves in which conditions they are general helix and slant helix. Finally, we exemplify them with figures.

The Moving-Frame Method for the Iterated-Integrals Signature: Orthogonal Invariants

Geometric, robust-to-noise features of curves in Euclidean space are of great interest for various applications such as machine learning and image analysis. We apply Fels–Olver’s moving-frame method (for geometric features) paired with the log-signature transform (for robust features) to construct a set of integral invariants under rigid motions for curves in {mathbb {R}}^d from the iterated-integrals signature. In particular, we show that one can algorithmically construct a set of invariants that characterize the equivalence class of the truncated iterated-integrals signature under orthogonal transformations, which yields a characterization of a curve in {mathbb {R}}^d under rigid motions (and tree-like extensions) and an explicit method to compare curves up to these transformations.

An inverse approach to hyperspheres of prescribed mean curvature in Euclidean space

Abstract We construct families of smooth functions H : ℝ n+1 → ℝ such that the Euclidean (n + 1)-space is completely filled by not necessarily round hyperspheres of mean curvature H at every point.

A New Variational Model for Shape Graph Registration with Partial Matching Constraints

This paper introduces a new extension of Riemannian elastic curve matching to a general class of geometric structures, which we call (weighted) shape graphs, that allows for shape registration with partial matching constraints and topological inconsistencies. Weighted shape graphs are the union of an arbitrary number of component curves in Euclidean space with potential connectivity constraints between some of their boundary points, together with a weight function defined on each component curve. The framework of higher-order invariant Sobolev metrics is particularly well suited for constructing notions of distances and geodesics between unparametrized curves. The main difficulty in adapting this framework to the setting of shape graphs is the absence of topological consistency, which typically results in an inadequate search for an exact matching between two shape graphs. We overcome this hurdle by defining an inexact variational formulation of the matching problem between (weighted) shape graphs of any underlying topology, relying on the convenient measure representation given by varifolds to relax the exact matching constraint. We then prove the existence of minimizers to this variational problem when we choose Sobolev metrics of sufficient regularity and a total variation (TV) regularization on the weight function. We propose a numerical optimization approach which adapts the smoothed fast iterative shrinkage-thresholding algorithm (SFISTA) to deal with $TV$ norm minimization and allows us to reduce the matching problem to solving a sequence of smooth unconstrained minimization problems. We finally illustrate the capabilities of our new model through several examples showcasing its ability to tackle partially observed and topologically varying data.

A New Method to Obtain PH-Helical Curves in E^(n+1)

Helical curves are constructed by the property that their unit tangents make a constant angle with a chosen constant direction. There are relations between polynomial planar curves, helices and Pythagorean-hodograph or shortly PH-curves. The aim of this paper is to obtain a method which generate PH-curves and PH-helical curves from a planar curve in Euclidean Space E^(n+1). Furthermore, some examples are given in E^4 and E^5 to explain the method neatly.

Characterizations of Framed Curves in Four-Dimensional Euclidean Space

Framed curves in Euclidean space are used to investigate singular curves and are important for singularity theory. In this study, framed curves in four-dimensional Euclidean space are introduced and new results are obtained. The relation of framed curves with Frenet curves in four-dimensional Euclidean space is given and Bishop-type frame of framed curves is introduced with the help of Euler angles. In addition, by using Bishop-type framed curves in four-dimensional Euclidean space, framed rectifying curves, framed osculating curves and framed normal curves are introduced. Also, some characterizations depending on framed curvatures are obtained.

Parabolic hypersurfaces with constant mean curvature in Euclidean space

In this paper we consider $O(m)\times O(n)$-invariant parabolic hypersurfaces of Euclidean space with constant mean curvature. We analyse the orbit space of the $O(m)\times O(n)$-group on $\mathbb{R}^{m+n}$ to give some classification results of these hypersurfaces for which the Gauss-Kronecker curvature does not change its sign.

Extrinsic Black Hole Uniqueness in Pure Lovelock Gravity

We define a notion of extrinsic black hole in pure Lovelock gravity of degree $k$ which captures the essential features of the so-called Lovelock-Schwarzschild solutions, viewed as rotationally invariant hypersurfaces with null $2k$-mean curvature in Euclidean space $\mathbb R^{n+1}$, $2\leq 2k\leq n-1$. We then combine a regularity argument with a rigidity result by Ara\'ujo-Leite to prove, under a natural ellipticity condition, a global uniqueness theorem for this class of black holes. As a consequence we obtain, in the context of the corresponding Penrose inequality for graphs established by Ge-Wang-Wu, a local rigidity result for the Lovelock-Schwarzschild solutions.

On the classification of framed rectifying curves in Euclidean space

There are many studies on regular rectifying curves in classical differential geometry, and important results have been obtained. However, these studies are limited for a smooth curve with singular points. To examine such curves and surfaces, the concept of framed curve, which is the general form of regular and Legendre curves, is used. Framed curves are defined as curves that have a moving frame with singular points in Euclidean space. We investigate framed rectifying curves via the dilation of framed curves on S2 in . Moreover, the result of dilation of framed curves is the framed rectifying curve or not. We give some classifications for the dilation of framed curves. Finally, we give some related examples with their figures.

Parallel Transport Frame of Smarandache Curves in Euclidean Space

In the present work, we introduce Parallel transport frames of Smarandache curves in Euclidean space. In the first section, we give the basic tools of a parallel transport frame of a curve in 4-dimensional Euclidean space. In the second section, we study Smarandache curve of Euclidean space in parallel transport frame, we solve a few theorems, corollaries and examples. Again third section, we define parallel transport frame to the Smarandache curve and obtain some definitions and their apparatus. Further fourth section, we have also explained to Frenet frame of principal normal, binomial and their derivatives in the curvatures of the curve. In the end section, we discussed about the Smarandache curve in the Euclidean space of all apparatus Frenet-Serret in the differential geometry.

Area minimizing surfaces of bounded genus in metric spaces

AbstractThe Plateau–Douglas problem asks to find an area minimizing surface of fixed or bounded genus spanning a given finite collection of Jordan curves in Euclidean space. In the present paper we solve this problem in the setting of proper metric spaces admitting a local quadratic isoperimetric inequality for curves. We moreover obtain continuity up to the boundary and interior Hölder regularity of solutions. Our results generalize corresponding results of Jost and Tomi-Tromba from the setting of Riemannian manifolds to that of proper metric spaces with a local quadratic isoperimetric inequality. The special case of a disc-type surface spanning a single Jordan curve corresponds to the classical problem of Plateau, in proper metric spaces recently solved by Lytchak and the second author.

Helical Extension Curve of a Space Curve

In this paper, we introduce a new class of curves that we call helical extension curve in Euclidean space. Then we investigate the differential geometric properties of the helical extension curves. The main result of the paper is that the helical extension curve of a helix is a plane curve. Moreover, we give several simple characterizations of helical extension curve of special curves such as spherical curve.