$h$-Principles for curves and knots of constant torsion

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.

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.

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.

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.

Smooth motion generation is an important issue in the computer animation and virtual reality (VR) area. In general, the motion of a rigid body consists of translation and orientation. The former is described by a space curve in 3-dimensional Euclidean space, while the latter is represented by a curve in the unit quaternion space. Although there are well-known techniques for smoothing the translation curve in the Euclidean space, few results have been reported for smoothing motion as a whole. This paper improves the previous study and provides a more robust algorithm, which seeks to minimize the weighted sum of the strain-energy and the sum of the squared errors.

The aim of this study is to examine the relations between Tzitzeica curves and Smarandache curves in Euclidean space. In addition, the necessary and sufficient conditions for Smarandache curves to be Tzitzeica curves in 3-dimensional Euclidean space are investigated and examples are given.

In this paper, we give some characterization for a osculating curve in 3-dimensional Euclidean space and we define a osculating curve in the Euclidean 4-space as a curve whose position vector always lies in orthogonal complement

This article studies complete -dimensional surfaces of nonpositive extrinsic 2-dimensional sectional curvature and nonpositive -dimensional curvature (for even) in Euclidean space , in the sphere , in the complex projective space , and in a Riemannian space . If the embedding codimension is sufficiently small, then a compact surface in or is a totally geodesic great sphere or complex projective space, respectively. If is a compact surface of negative extrinsic 2-dimensional curvature in a Riemannian space , then there are restrictions on the topological type of the surface. It is shown that a compact Riemannian manifold of nonpositive -dimensional curvature cannot be isometrically immersed as a surface of small codimension. The order of growth of the volume of complete noncompact surfaces of nonpositive -dimensional curvature in Euclidean space is estimated; it is determined when such surfaces are cylinders. A question about surfaces in which are homeomorphic to a sphere and which have nonpositive extrinsic curvature is looked at.Bibliography: 25 titles.

To any metric space it is possible to associate the cardinal invariant corresponding to the least number of rectifiable curves in the space whose union is not meagre. It is shown that this invariant can vary with the metric space considered, even when restricted to the class of convex subspaces of separable Banach spaces. As a corollary it is obtained that it is consistent with set theory that any set of reals of size ℵ1 is meagre yet there are ℵ1 rectifiable curves in ℝ3 whose union is not meagre. The consistency of this statement when the phrase “rectifiable curves” is replaced by “straight lines” remains open.

Homogeneous matrices are widely used to represent geometric transformations in computer graphics, with interpolation between those matrices being of high interest for computer animation. Many approaches have been proposed to address this problem, including computing matrix curves from curves in Euclidean space by registration, representing one-parameter curves on manifold by rational representations, changing subdivisional methods generating curves in Euclidean space to corresponding methods working for matrix curve generation, and variational methods. In this paper, we propose a scheme to generate rational one-parameter matrix curves based on exponential map for interpolation, and demonstrate how to obtain higher smoothness from existing curves. We also give an iterative technique for rapid computing of these curves. We take the computation as solving an ordinary differential equation on manifold numerically by a generalized Euler method. Furthermore, we give this algorithm’s bound of the error and prove that the bound is proportional to the shift length when the shift length is sufficiently small. Compared to direct computation of the matrix functions, our Euler solution is faster.

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.

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.

Comparing curves in homogeneous spaces

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 $$\frac{1}{2|H|}$$ .

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.