Curves In Euclidean Space Research Articles

Products of Euclidean Metrics, Applied to Proximity Problems among Curves

Approximate Nearest Neighbor (ANN) search is a fundamental computational problem that has benefited from significant progress in the past couple of decades. However, most work has been devoted to pointsets, whereas complex shapes have not been sufficiently addressed. Here, we focus on distance functions between discretized curves in Euclidean space: They appear in a wide range of applications, from road segments and molecular backbones to time-series in general dimension. For ℓ p -products of Euclidean metrics, for any constant p , we propose simple and efficient data structures for ANN based on randomized projections: These data structures are of independent interest. Furthermore, they serve to solve proximity questions under a notion of distance between discretized curves, which generalizes both discrete Fréchet and Dynamic Time Warping distance functions. These are two very popular and practical approaches to comparing such curves. We offer, for both approaches, the first data structures and query algorithms for ANN with arbitrarily good approximation factor, at the expense of increasing space usage and preprocessing time over existing methods. Query time complexity is comparable or significantly improved by our methods; our algorithm is especially efficient when the length of the curves is bounded. Finally, we focus on discrete Fréchet distance when the ambient space is high dimensional and derive complexity bounds in terms of doubling dimension as well as an improved approximate near neighbor search.

Bertrand and Mannheim curves of framed curves in the3-dimensional Euclidean space

A Bertrand curve is a space curve whose principal normal line is the same as the principal normal line of another curve. On the other hand, a Mannheim curve is a space curve whose principal normal line is the same as the binormal line of another curve. By definitions, another curve is a parallel curve with respect to the direction of the principal normal vector. Even if that is the regular case, the existence conditions of the Bertrand and Mannheim curves seem to be wrong in some previous research. Moreover, parallel curves may have singular points. As smooth curves with singular points, we consider framed curves in the Euclidean space. Then we define and investigate Bertrand and Mannheim curves of framed curves. We clarify that the Bertrand and Mannheim curves of framed curves are dependent on the moving frame.

Asymptotics of mean value points: a survey

This expository article is devoted to some results about non-trivial asymptotical estimates of mean value points in analytical and geometrical theorems. In particular, we provide detailed proofs of the first and the second Ionin’s conjectures, respectively concerning with the first mean value theorem for integrals and the Taylor formula with the Lagrange remainder. We consider also some related results on divided differences and the multi-dimensional mean value theorem for integrals. In the final section, we deal with the asymptotic behavior of support points for planar continuous curves and tangent points for differentiable curves in Euclidean spaces.

Elastic null curve flows, nonlinear C-integrable systems, and geometric realization of Cole-Hopf transformations

Elastic (stretching) flows of null curves are studied in three-dimensional Minkowski space. As a main tool, a natural type of moving frame for null curves is introduced, without use of the pseudo-arclength. This new frame is related to a Frenet null frame by a gauge transformation that belongs to the little group contained in the Lorentz group SO(2, 1) and provides an analog of the Hasimoto transformation (relating a parallel frame to a Frenet frame for curves in Euclidean space). The Cartan structure equations of the transformed frame are shown to encode a hereditary recursion operator giving a two-component generalization of the recursion operator of Burgers equation, as well as a generalization of the Cole-Hopf transformation. Three different hierarchies of integrable systems are obtained from the various symmetries of this recursion operator. The first hierarchy contains two-component Burgers-type and nonlinear Airy-type systems; the second hierarchy contains novel quasilinear Schrödinger-type (NLS) systems; and the third hierarchy contains semilinear wave equations (in two-component system form). Each of these integrable systems is shown to correspond to a geometrical flow of a family of elastic null curves in three-dimensional Minkowski space.

Some New Characterizations of Symplectic Curve in 4-Dimensional Symplectic Space

It is well known that there exist characterizations for curve in Euclidean space.$\ $Also, a lot of authors extended this characterizations for Minkowski space and obtained very different results. In this paper, we introduce the geometric properties of Symplectic Curve in $% 4$-Dimensional Symplectic Space which given by \cite{7,10}. Later we obtained the conditions for Symplectic Curve to lie on some subspaces of $% 4$-Dimensional Symplectic Space and we give some characterizations and theorems for these curves.

Discrete Cosserat Rod Kinematics Constructed on the Basis of the Difference Geometry of Framed Curves—Part I: Discrete Cosserat Curves on a Staggered Grid

The theory of Cosserat rods provides versatile models to simulate large spatial deformations of slender flexible structures. As the strain measures of the mechanical theory are given in terms of the differential invariants of Cosserat curves, the kinematics of Cosserat rods is closely related to the differential geometry of framed curves. We utilize ideas from the difference geometry of framed curves in Euclidean space to construct the discrete kinematics of Cosserat rod models on a staggered grid, preserving the essential geometric properties independent of the coarseness of the discretization.

Translation surfaces with non-zero constant mean curvature in Euclidean space

We prove that the only surface in 3-dimensional Euclidean space $${\mathbb {R}}^3$$ with constant and non-zero mean curvature H, constructed by the sum of a planar curve and a space curve, is the circular cylinder of radius $$\frac{1}{2|H|}$$ .

Application of the geometry of curves in Euclidean space

In this article the so called induced spin velocities are studied, and it is an improvement of the paper [13] using the geometry of curves in 3-dimensional Euclidean space. Some essential properties of them are given, and they are rather different than the ordinary velocities. Indeed, the induced spin velocities are non-inertial and instead of the Lorentz transformations for them the Galilean transformations should be used. The induced spin velocity is derived in terms of the curvature and torsion of the trajectory. Two applications of the induced spin velocities are studied.

On minimal surfaces with the same asymptotic curve in Euclidean space

On minimal surfaces with the same asymptotic curve in Euclidean space

Comparing curves in homogeneous spaces

Of concern is the study of the space of curves in homogeneous spaces. Motivated by applications in shape analysis we identify two curves if they only differ by their parametrization and/or a rigid motion. For curves in Euclidean space the Square-Root-Velocity-Function (SRVF) allows to define and efficiently compute a distance on this infinite dimensional quotient space. In this article we present a generalization of the SRVF to curves in homogeneous spaces. We prove that, under mild conditions on the curves, there always exist optimal reparametrizations realizing the quotient distance and demonstrate the efficiency of our framework in selected numerical examples.

General criteria for curves to be simple

We extend previous results for parametrized curves in euclidean space to be simple. The new condition depends as before on Ahlfors' Schwarzian and considers a conformal metric on a given interval and the new diameter. We derive some applications, among which we find Becker type conditions that depend on a pre-Schwarzian.

Remarks on hypersurfaces with constant higher order mean curvature in Euclidean space

In this paper we consider complete oriented hypersurfaces of Euclidean space with constant higher order mean curvature and having two principal curvatures, one of them simple. As an application of the so called principal curvature theorem, a purely geometric result on the principal curvatures of the hypersurface given by Smyth and Xavier (Invent Math 90:443–450, 1987), we characterize those hypersurfaces for which the Gauss–Kronecker curvature does not change sign, extending to the general n-dimensional case a previous result for surfaces due to Klotz and Osserman.

Characterization of Curves that Lie on a Geodesic Sphere or on a Totally Geodesic Hypersurface in a Hyperbolic Space or in a Sphere

The consideration of the so-called rotation minimizing frames allows for a simple and elegant characterization of plane and spherical curves in Euclidean space via a linear equation relating the coefficients that dictate the frame motion. In this work, we extend these investigations to characterize curves that lie on a geodesic sphere or totally geodesic hypersurface in a Riemannian manifold of constant curvature. Using that geodesic spherical curves are normal curves, i.e., they are the image of an Euclidean spherical curve under the exponential map, we are able to characterize geodesic spherical curves in hyperbolic spaces and spheres through a non-homogeneous linear equation. Finally, we also show that curves on totally geodesic hypersurfaces, which play the role of hyperplanes in Riemannian geometry, should be characterized by a homogeneous linear equation. In short, our results give interesting and significant similarities between hyperbolic, spherical, and Euclidean geometries.

Biharmonic hypersurfaces with constant scalar curvature in space forms

Let $M^n$ be a biharmonic hypersurface with constant scalar curvature in a space form $\mathbb M^{n+1}(c)$. We show that $M^n$ has constant mean curvature if $c>0$ and $M^n$ is minimal if $c\leq0$, provided that the number of distinct principal curvatures is no more than 6. This partially confirms Chen's conjecture and Generalized Chen's conjecture. As a consequence, we prove that there exist no proper biharmonic hypersurfaces with constant scalar curvature in Euclidean space $\mathbb E^{n+1}$ or hyperbolic space $\mathbb H^{n+1}$ for $n<7$.

Tangents, rectifiability, and corkscrew domains

In a recent paper, Csornyei and Wilson prove that curves in Euclidean space of σ-finite length have tangents on a set of positive H 1-measure. They also show that a higher dimensional analogue of this result is not possible without some additional assumptions. In this note, we show that if Σ ⊆ Rd+1 has the property that each ball centered on Σ contains two large balls in different components of Σc and Σ has σ-finite H d-measure, then it has d-dimensional tangent points in a set of positive H d-measure. As an application, we show that if the dimension of harmonic measure for an NTA domain in Rd+1 is less than d, then the boundary domain does not have σ-finite H d-measure. We also give shorter proofs that Semmes surfaces are uniformly rectifiable and, if Ω ⊆ Rd+1 is an exterior corkscrew domain whose boundary has locally finite H d-measure, one can find a Lipschitz subdomain intersecting a large portion of the boundary.

Generalized Fermi–Walker derivative and non-rotating frame

In this paper, generalized Fermi–Walker derivative, generalized Fermi–Walker parallelism and generalized non-rotating frame concepts are given for Frenet frame, Darboux frame and Bishop frame for any curve in Euclidean space. Being generalized, non-rotating frame conditions are analyzed for each frames along the curve. Then we show that Frenet and Darboux frames are generalized non-rotating frames along all curves and also Bishop frame is generalized non-rotating frame along planar curves in Euclidean space.

Smooth orientation interpolation using parametric quintic-polynomial-based quaternion spline curve

In this paper, a G 2 continuous quintic-polynomial-based unit quaternion interpolation spline curve with tension parameters is presented to interpolate a given sequence of solid orientations. The curve in unit quaternion space S 3 is an extension of the quintic polynomial interpolation spline curve in Euclidean space. It preserves the interpolatory property and G 2 continuity. Meanwhile, the unit quaternion interpolation spline curve possesses the local shape adjustability due to the presence of tension parameters. The change of one tension parameter will only affect the adjacent two pieces of curves. Compared with the traditional B-spline unit quaternion interpolation curve and v -spline unit quaternion interpolation curve, the proposed curve can automatically interpolate the given data points, without solving the nonlinear system of equations over quaternions to obtain the control points, which greatly improves the computational efficiency. Simulation results demonstrate the effectiveness of the proposed scheme.

Smooth interpolation of key frames in a Riemannian shell space

Splines and subdivision curves are flexible tools in the design and manipulation of curves in Euclidean space. In this paper we study generalizations of interpolating splines and subdivision schemes to the Riemannian manifold of shell surfaces in which the associated metric measures both bending and membrane distortion. The shells under consideration are assumed to be represented by Loop subdivision surfaces. This enables the animation of shells via the smooth interpolation of a given set of key frame control meshes. Using a variational time discretization of geodesics efficient numerical implementations can be derived. These are based on discrete geodesic interpolation, a discrete geometric logarithm, a discrete exponential map, and discrete parallel transport. With these building blocks at hand discrete Riemannian cardinal splines and three different types of discrete, interpolatory subdivision schemes are defined. Numerical results for two different subdivision shell models underline the potential of this approach in key frame animation.

Non-closed curves in ℝn with finite total first curvature arising from the solutions of an ODE

The solution space of a constant coefficient ODE gives rise to a natural real analytic curve in Euclidean space. We give necessary and sufficient conditions on the ODE to ensure that this curve is a proper embedding of infinite length or has finite total first curvature. If all the roots of the associated characteristic polynomial are simple, we give a uniform upper bound for the total first curvature and show the optimal uniform upper bound must grow at least linearly with the order $n$ of the ODE. We then examine the case where multiple roots are permitted. We present several examples illustrating that a curve can have finite total first curvature for positive/negative time and infinite total first curvature for negative/positive time as well as illustrating that other possibilities may occur.

Geometric inequalities and rigidity theorems on equatorial spheres

We prove rigidity for hypersurfaces with boundary in the unit $(n+1)$-sphere with scalar curvature bounded below by $n(n-1)$. Under appropriate boundary conditions, the hypersurfaces are shown to be part of the equatorial spheres. The lower bound $n(n-1)$ is critical in the sense that the hypersurface may contain geodesic points and some natural differential operators are fully degenerate at geodesic points. We overcome the difficulty by studying the geometry of level sets of a height function, via new geometric inequalities. Some rigidity results of hyperplanes and generalized cylinders are also obtained for hypersurfaces with boundary and with nonnegative scalar curvature in Euclidean space.