Euclidean Curves Research Articles

Osculating ruled surfaces of a surface along a curve in Euclidean 3-space E^3

Osculating ruled surfaces of a surface along a curve in Euclidean 3-space E^3

New version of Bäcklund transformations in Euclidean 3‐space

In this paper, we study the Bäcklund transformations for the adjoint curve in the Euclidean 3‐space. Firstly, it is obtained some essential equations of the Bäcklund transformation. After this, we give a new theorem, the Bäcklund transformations for the adjoint curve in Euclidean 3‐space.

A new aspect of rectifying curves and ruled surfaces in Galilean 3-space

A curve is named as rectifying curve if its position vector always lies in its rectifying plane. There are lots of papers about rectifying curves in Euclidean and Minkowski spaces. In this paper, we give some relations between extended rectifying curves and their modified Darboux vector fields in Galilean 3-Space. The other aim of the paper is to introduce the ruled surfaces whose base curve is rectifying curve. Further, we prove that the parameter curve of the surface is a geodesic.

Smarandache curves in Euclidean 4- space E 4

Smarandache curves in Euclidean 4- space E 4

Smarandache curves in Euclidean 4- space E 4

Smarandache curves in Euclidean 4- space E 4

Smarandache curves in Euclidean 4- space E 4

The purpose of this paper is to study Smarandache curves in the 4-dimensional Euclidean space E4, and to obtain the Frenet–Serret and Bishop invariants for the Smarandache curves in E4. The first, the second and the third curvatures of Smarandache curves are calculated. These values depending upon the first, the second and the third curvature of the given curve.

Vessel tree extraction using radius-lifted keypoints searching scheme and anisotropic fast marching method

Geodesic methods have been widely applied to image analysis. They are particularly efficient to extract a tubular structure, such as a blood vessel, given its two endpoints in a 2D or 3D medical image. We address here a more difficult problem: the extraction of a full vessel tree structure given a single initial root point, by growing a collection of keypoints or new initial source points, connected by minimal geodesic paths. In this article, those keypoints are iteratively added, using a new detection criteria, which utilize the weighted geodesic distances with respect to a radius-lifted Riemannian metric, the standard Euclidean curve length and a path score. Two main weaknesses of classical keypoints searching approach are that the weighted geodesic distance and the Euclidean path length do not take into account the orientation of the tubular structure or object boundaries, due to the use of an isotropic geodesic Riemannian metric, and suffer from a leakage problem. In contrast, we use an anisotropic geodesic Riemannian metric, and develop new criteria for selecting keypoints based on the path score and automatically stopping the tree growth. Experimental results demonstrate that our method can obtain the expected results, which can extract vessel structures at a finer scale, with increased accuracy.

Some characterizations of constant breadth curves in Euclidean 4-space

Some characterizations of constant breadth curves in Euclidean 4-space

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.

BOUNDARY DIMENSIONS OF FRACTAL TILINGS

A fractal tiling is a tiling which possesses self-similarity and the boundary of which is a fractal. In this paper, we investigate the boundary dimension of [Formula: see text]- and [Formula: see text]-tilings. We first derive an explicit recursion formula for the boundary edges of [Formula: see text]- and [Formula: see text]-tilings. Then we present an analytical expression for their fractal boundary dimensions using matrix methods. Results indicate that, as [Formula: see text] increases, the boundaries of [Formula: see text]- and [Formula: see text]-tilings will degenerate into general Euclidean curves. The method proposed in this paper can be extended to compute the boundary dimensions of other kinds of fractal tilings.

The classical Korteweg capillarity system: geometry and invariant transformations

A class of invariant transformations is presented for the classical Korteweg capillarity system. The invariance is an extension of a kind originally introduced in an anisentropic gasdynamics context. In a particular instance, application of the invariant transformation leads to a deformed one-parameter class of Kármán–Tsien-type capillarity laws associated with a deformation of an integrable nonlinear Schrödinger-type equation which incorporates a de Broglie–Bohm potential. The latter and another integrable case associated with the classical Boussinesq equation may be linked to the motion of curves in Euclidean and projective space so that both the invariant transformation and the Galilean invariance of the capillarity system may be interpreted in a geometric and soliton-theoretic manner. The work is set in the broader context of other connections of invariant transformations in gasdynamics with soliton theory.

Generalized Mannheim quaternionic curves in Euclidean 4-space

In this paper, we define a generalized Mannheim quaternionic curve in the four-dimensional Euclidean space R 4 and investigate the properties of it.

Characterization of curves in E2n+1 with 1-type Darboux vector

In this study, we give some characterizations on the Darboux instantaneous rotation vector field of the curves in Euclidean (2n + 1)-space E2n+1 by using Laplacian operator. Further, we give necessary and sufficient conditions for unit speed space curves to have 1-type Darboux vector.

The behavior of curvature functions at cusps and inflection points

At a 3/2-cusp of a given plane curve γ(t), both of the Euclidean curvature κg and the affine curvature κA diverge. In this paper, we show that each of |sg|κg and (sA)2κA (called the Euclidean and affine normalized curvature, respectively) at a 3/2-cusp is a C∞-function of the variable t, where sg (resp. sA) is the Euclidean (resp. affine) arclength parameter of the curve corresponding to the 3/2-cusp sg=0 (resp. sA=0). Moreover, we give a characterization of the behavior of the curvature functions κg and κA at 3/2-cusps. On the other hand, inflection points are also singular points of curves in affine geometry. We give a similar characterization of affine curvature functions near generic inflection points. As an application, new affine invariants of 3/2-cusps and generic inflection points are given.

On the total curvature of curves in a Minkowski space

We consider simple closed curves in a Minkowski space. We give bounds of the total Minkowski curvature of the curve in terms of the total Euclidean curvature and of normal curvatures on the indicatrix (supposed to be a central symmetric hypersurface) of the Minkowski norm. Corollaries of this result provide analogues to Fenchel and Fary-Milnor theorems. We also give an upper bound of the Minkowski length of a simple closed curve contained in a Minkowski ball of radius R, in terms of the total Minkowski curvature and of normal curvatures on the indicatrix. Whenever the Minkowski space is Euclidean our results reduce to the classical ones.

A refinement of Foreman’s four-vertex theorem and its dual version

A strictly convex curve is a C∞-regular simple closed curve whose Euclidean curvature function is positive. Fix a strictly convex curve Γ, and take two distinct tangent lines l1 and l2 of Γ. A few years ago, Brendan Foreman proved an interesting four-vertex theorem on semiosculating conics of Γ, which are tangent to l1 and l2, as a corollary of Ghys’s theorem on diffeomorphisms of S1. In this paper, we prove a refinement of Foreman’s result. We then prove a projectively dual version of our refinement, which is a claim about semiosculating conics passing through two fixed points on Γ. We also show that the dual version of Foreman’s four-vertex theorem is almost equivalent to the Ghys’s theorem.

New open string solutions inAdS5

We describe new solutions for open string moving in AdS_5 and ending in the boundary, namely dual to Wilson loops in N=4 SYM theory. First we introduce an ansatz for Euclidean curves whose shape contains an arbitrary function. They are BPS and the dual surfaces can be found exactly. After an inversion they become closed Wilson loops whose expectation value is $W=\exp(-\sqrt{\lambda})$. After that we consider several Wilson loops for N=4 SYM in a pp-wave metric and find the dual surfaces in an AdS_5 pp-wave background. Using the fact that the pp-wave is conformally flat, we apply a conformal transformation to obtain novel surfaces describing strings moving in AdS space in Poincare coordinates and dual to Wilson loops for N=4 SYM in flat space.

Curve Shortening and the Rendezvous Problem for Mobile Autonomous Robots

If a smooth, closed, and embedded curve is deformed along its normal vector field at a rate proportional to its curvature, it shrinks to a circular point. This curve evolution is called Euclidean curve shortening and the result is known as the Gage-Hamilton-Grayson theorem. Motivated by the rendezvous problem for mobile autonomous robots, we address the problem of creating a polygon shortening flow. A linear scheme is proposed that exhibits several analogues to Euclidean curve shortening: The polygon shrinks to an elliptical point, convex polygons remain convex, and the perimeter of the polygon is monotonically decreasing.

Eigenvalue Equations in Curve Theory Part II: Evolutes and Involutes

We study plane Euclidean curves with positive curvature κ and spherical parametrization s.t. their radius of curvature κ−1 satisfies an eigenvalue equation. We investigate this class in detail in terms of evolutes and involutes and their geometric properties in relation to the eigenvalue equations considered. For a curve c in this class, the radius of curvature of its evolute satisfies the same eigenvalue equation like κ−1 of c, while the radii of curvature of its involutes satisfy a slightly modified eigenvalue equation. We give a complete local classification of all types of curves in this class.

Number Counts at 3 μm < λ < 10 μm from the Spitzer Space Telescope

Infrared source counts at wavelengths 3 μm < λ < 10 μm cover more than 10 mag in source brightness, reach 4 orders of magnitude in surface density, and reach an integrated surface density of 105 sources deg-2. At m < 14 mag, most of the sources are Galactic stars, in agreement with models. After removal of Galactic stars, galaxy counts are consistent with what few measurements exist at nearby wavelengths. At 3.6 and 4.5 μm, the galaxy counts follow the expectations of a Euclidean world model down to ∼16 mag and drop below the Euclidean curve for fainter magnitudes. Counts at these wavelengths begin to show decreasing completeness around 19.5 mag. At 5.8 and 8 μm, the counts relative to a Euclidean world model show a large excess at bright magnitudes. This is probably because local galaxies emit strongly in the aromatic dust ("polycyclic aromatic hydrocarbon") features. The counts at 3.6 μm resolve less than 50% of the cosmic infrared background at that wavelength.