Curves In Euclidean Space Research Articles

Constant Ratio Quaternionic Curves in Euclidean Spaces

In this paper, we give some characterizations of quaternionic curves whose position vectors can be written as linear combinations of their Serret-Frenet vectors in Euclidean 3-space \({\mathbb{E}^{3}}\) and Euclidean 4-space \({\mathbb{E}^{4}}\). We characterize such curves in terms of their curvature functions. Finally, we give the necessary and sufficient conditions for these types of curves to become constant ratio, T-constant, and N-constant.

Stable hypersurfaces with zero scalar curvature in Euclidean space

In this paper we prove some results concerning stability of hypersurfaces in the four dimensional Euclidean space with zero scalar curvature. First we prove there is no complete stable hypersurface with zero scalar curvature, polynomial growth of integral of the mean curvature, and with the Gauss-Kronecker curvature bounded away from zero. We conclude this paper giving a sufficient condition for a regular domain to be stable in terms of the mean and the Gauss-Kronecker curvatures of the hypersurface and the radius of the smallest extrinsic ball which contains the domain.

Quaternionic osculating curves in Euclidean and semi-Euclidean space

In this study, the osculating curves in Euclidean space E3 and E4, well known in differential geometry, are studied through the instrumentality of quaternions. We inoculate sundry delineations for quaternionic osculating curves in the Euclidean space E3, then we portray the quaternionic osculating curve in E4 as a quaternionic curve whose position vector every time reclines in the orthogonal complement N½ (or N⅓) of its first binormal vector field N2 (or N3), where {T,N1,N2,N3} be the Frenet instrumentations of the quaternionic curve in the Euclidean space E4. We feature quaternionic oscu­lating curves from the point of view their curvature functions K, k and (r — K) and serve the necessary and the sufficient conditions for arbitrary quaternionic curve in E4 to be a quaternionic osculating. Moreover, we gain an explicit equation of a quaternionic osculating curve in E4. In the last two section, we described quaternionic osculating curves in the semi-Euclidean space and some theorems are testified.

Hypersurfaces with constant higher order mean curvature in Euclidean space

In this paper we derive sharp estimates for the infimum and for the supremum of the squared norm of the second fundamental form of complete oriented hypersurfaces of Euclidean space with constant higher order mean curvature and having two principal curvatures, one of them simple. Besides, we characterize those hypersurfaces for which any of these bounds is attained. Our results will be an application of a purely geometric result on the principal curvatures of the hypersurface, the so called principal curvature theorem, given by Smyth and Xavier (Invent Math 90:443–450, 1987).

Uniform Lxp–Lx,rq improving for dilated averages over polynomial curves

Numerous authors have considered the problem of determining the Lebesgue space mapping properties of the operator A given by convolution with affine arc-length measure on some polynomial curve in Euclidean space. Essentially, A takes weighted averages over translates of the curve. In this paper a variant of this problem is discussed where averages over both translates and dilates of a fixed curve are considered. The sharp range of estimates for the resulting operator is obtained in all dimensions, except for an endpoint. The techniques used are redolent of those previously applied in the study of A . In particular, the arguments are based upon the refinement method of Christ, although a significant adaptation of this method is required to fully understand the additional smoothing afforded by averaging over dilates.

A problem of integral geometry for a family of curves with incomplete data

The unknown boundary problem of integral geometry is considered. Known data are integrals of unknown functions over an unknown family of curves in Euclidean space of any dimension. It is mentioned that, under the traditional approach, the integral geometry problem under consideration would be underdetermined. The sought object is the discontinuity surfaces of the integrands. A uniqueness theorem for the solution of the problem specified above is proved.

Generation of spherical non-uniform rational basis spline curves and its application in five-axis machining

A method of generating spherical non-uniform rational basis spline curves based on De Boor’s algorithm is presented in this article. The spherical curve preserves many good properties from non-uniform rational basis spline curves in Euclidean space, such as local modification property, convex hull property, rotation invariant property, knot insertion property, and so on. Construction of closed spherical non-uniform rational basis spline curve will be discussed too. Furthermore, a progressive iterative approximation scheme is proposed to approximate the given points on unit sphere using spherical non-uniform rational basis spline curves. The convergence and curve continuity will be discussed. Since the analytical expression of the spherical non-uniform rational basis spline curve is messy, numerical differentiation is used to test its continuity at interior knots. Finally, several examples are presented to verify the effectiveness of the proposed method. The proposed method is applied on tool path generation of five-axis machining to avoid interference by adjusting fewer directions and obtain smooth tool path simultaneously.

CURVE COUPLES AND SPACELIKE FRENET PLANES IN MINKOWSKI 3-SPACE

Abstract. In this study, we have investigated the possibility ofwhether any spacelike Frenet plane of a given space curve in Minkowski3-space E 31 also is any spacelike Frenet plane of another space curvein the same space. We have obtained some characterizations of agiven space curve by considering nine possible case. 1. IntroductionIn the theory of curves in Euclidean space, one of the importantand interesting problem is characterization of a regular curve. In thesolution of the problem, the curvature functions k 1 (or {) and k 2 (or ˝)of a regular curve have an e ective role. For example: if k 1 = 0 = k 2 ,then the curve is a geodesic or if k 1 = constant 6= 0 and k 2 = 0;thenthe curve is a circle with radius (1=k 1 ), etc. Thus we can determine theshape and size of a regular curve by using its curvatures.Another way in the solution of the problem is the relationship betweenthe Frenet vectors of the curves (see [6]). For instance Bertrand curves:In 1845, Saint Venant (see [6] and [8]) proposed the question whetherupon the surface generated by the principal normal of a curve, a secondcurve can exist which has for its principal normal the principal normalof the given curve. This question was answered by Bertrand in 1850 ina paper which he showed that a necessary and sucient condition forthe existence of such a second curve is that a linear relationship withconstant coecients shall exist between the rst and second curvaturesof the given original curve. In other word, if we denote rst and second

A distance on curves modulo rigid transformations

We propose a geometric method for quantifying the difference between parametrized curves in Euclidean space by introducing a distance function on the space of parametrized curves up to rigid transformations (rotations and translations). Given two curves, the distance between them is defined as the infimum of an energy functional which, roughly speaking, measures the extent to which the jet field of the first curve needs to be rotated to match up with the jet field of the second curve. We show that this energy functional attains a global minimum on the appropriate function space, and we derive a set of first-order ODEs for the minimizer.

Magnetic Effects in Curved Quantum Waveguides

The interplay among the spectrum, geometry and magnetic field in tubular neighbourhoods of curves in Euclidean spaces is investigated in the limit when the cross section shrinks to a point. Proving a norm resolvent convergence, we derive effective, lower-dimensional models which depend on the intensity of the magnetic field and curvatures. The results are used to establish complete asymptotic expansions for eigenvalues. Spectral stability properties based on Hardy-type inequalities induced by magnetic fields are also analysed.

MANNHEIM CURVES IN 3-DIMENSIONAL SPACE FORMS

We define the Mannheim curve in a 3-dimensional Riemannian manifold, which is a generalization of the Mannheim curve in Euclidean space. In particular, we study the Mannheim curves and their partner curves in 3-dimensional space forms.

Erratum to “Some geometric properties of hypersurfaces with constant 𝑟-mean curvature in Euclidean space”

An erratum to the paper [D. Impera, L. Mari, and M. Rigoli, Some geometric properties of hypersurfaces with constant r r -mean curvature in Euclidean space, Proc. Amer. Math. Soc. 139 (2011), no. 6, 2207-2215] is presented.

Integrable Deformations of the (2+1)-Dimensional Heisenberg Ferromagnetic Model

Based on the covariant prolongation structure technique, we construct the integrable higher-order deformations of the (2+1)-dimensional Heisenberg ferromagnet model and obtain their su(2) × R(λ) prolongation structures. By associating these deformed multidimensional Heisenberg ferromagnet models with the moving space curve in Euclidean space and using the Hasimoto function, we derive their geometrical equivalent counterparts, i.e., higher-order (2+1)-dimensional nonlinear Schrödinger equations.

Pythagorean-hodograph curves in Euclidean spaces of dimension greater than 3

A polynomial Pythagorean-hodograph (PH) curve r(t)=(x1(t),…,xn(t)) in Rn is characterized by the property that its derivative components satisfy the Pythagorean condition x1′2(t)+⋯+xn′2(t)=σ2(t) for some polynomial σ(t), ensuring that the arc length s(t)=∫σ(t)dt is simply a polynomial in the curve parameter t. PH curves have thus far been extensively studied in R2 and R3, by means of the complex-number and the quaternion or Hopf map representations, and the basic theory and algorithms for their practical construction and analysis are currently well-developed. However, the case of PH curves in Rn for n>3 remains largely unexplored, due to difficulties with the characterization of Pythagorean (n+1)-tuples when n>3. Invoking recent results from number theory, we characterize the structure of PH curves in dimensions n=5 and n=9, and investigate some of their properties.

A2-singularities of hypersurfaces with non-negative sectional curvature in Euclidean space

In a previous work, the authors gave a definition of `front bundles'. Using this, we give a realization theorem for wave fronts in space forms, like as in the fundamental theorem of surface theory. As an application, we investigate the behavior of principal singular curvatures along A2-singularities of hypersurfaces with non-negative sectional curvature in Euclidean space.

On Submanifolds with Negative Curvature in Euclidean Space

In this paper we give a survey of results of different authors, among them Tenenblat, on the behaviour of submanifolds with negative curvature.

Injectivity Criteria for Holomorphic Curves in C<sup>n</sup>

Combining the definition of Schwarzian derivative for conformal mappings be- tween Riemannian manifolds given by Osgood and Stowe with that for parametrized curves in Euclidean space given by Ahlfors, we establish injectivity criteria for holomorphic curves � : D → C n . The result can be considered a generalization of a classical condition for univalence of Nehari.

Some geometric properties of hypersurfaces with constant $r$-mean curvature in Euclidean space

Let f: M → R m+1 be an isometrically immersed hypersurface. In this paper, we exploit recent results due to the authors to analyze the stability of the differential operator L r associated with the rth Newton tensor of f. This appears in the Jacobi operator for the variational problem of minimizing the r-mean curvature H r . Two natural applications are found. The first one ensures that under a mild condition on the integral of H r over geodesic spheres, the Gauss map meets each equator of S m infinitely many times. The second one deals with hypersurfaces with zero (r + 1)-mean curvature. Under similar growth assumptions, we prove that the affine tangent spaces f * T p M, p ∈ M, fill the whole R m+1 .

Shape Analysis of Elastic Curves in Euclidean Spaces

This paper introduces a square-root velocity (SRV) representation for analyzing shapes of curves in euclidean spaces under an elastic metric. In this SRV representation, the elastic metric simplifies to the IL(2) metric, the reparameterization group acts by isometries, and the space of unit length curves becomes the unit sphere. The shape space of closed curves is the quotient space of (a submanifold of) the unit sphere, modulo rotation, and reparameterization groups, and we find geodesics in that space using a path straightening approach. These geodesics and geodesic distances provide a framework for optimally matching, deforming, and comparing shapes. These ideas are demonstrated using: 1) shape analysis of cylindrical helices for studying protein structure, 2) shape analysis of facial curves for recognizing faces, 3) a wrapped probability distribution for capturing shapes of planar closed curves, and 4) parallel transport of deformations for predicting shapes from novel poses.

Integral conditions on the Schwarzian for curves to be simple or unknotted

By considering integral bounds on the Schwarzian derivative we extend previous results on sufficient conditions for curves in euclidean spaces to be simple or unknotted. The conditions are optimal.