Construction of interpolating space curves with arbitrary degree of geometric continuity

Geometric Hermite curves with minimum strain energy

The concept of the natural mate and the conjugate curves associated to a smooth curve in Euclidian 3-space were introduced initially by Dashmukh and others. In this paper, we give some extra results that add more properties of the natural mate and the conjugate curves associated with a smooth space curve in E 3 . The position vectors of the natural mate and the conjugate of a given smooth curve are investigated. Also, using the position vector of the natural mate, the necessary and sufficient condition for a smooth given curve to be a Bertrand curve is introduced. Moreover, a new characterization of a general helix is introduced.

In this article, we show that it is of frequent occurrence that smooth curves do have multiple components for some of their schemes of special divisors. For a smooth space curve C of degree d, we give sufficient conditions implying that has a multiple component, and we prove the existence of many space curves satisfying those conditions. As an example of a more general result for curves in ℙ r , we prove that general complete intersection curves of degree d in ℙ4 do have multiple components for the schemes and .

In this paper, we present an efficient method for the numerical investigation of three-dimensional non-integer-order convection-diffusion equation (CDE) based on radial basis functions (RBFs) in localized form and Laplace transform (LT). In our numerical scheme, first we transform the given problem into Laplace space using Laplace transform. Then, the local radial basis function (LRBF) method is employed to approximate the solution of the transformed problem. Finally, we represent the solution as an integral along a smooth curve in the complex left half plane. The integral is then evaluated to high accuracy by a quadrature rule. The Laplace transform is used to avoid the classical time marching procedure. The radial basis functions are important tools for scattered data interpolation and for solving partial differential equations (PDEs) of integer and non-integer order. The LRBF and global radial basis function (GRBF) are used to produce sparse collocation matrices which resolve the issue of the sensitivity of shape parameter and ill conditioning of system matrices and reduce the computational cost. The application of Laplace transformation often leads to the solution in complex plane which cannot be generally inverted. In this work, improved Talbot’s method is utilized which is an efficient method for the numerical inversion of Laplace transform. The stability and convergence of the method are discussed. Two test problems are considered to validate the numerical scheme. The numerical results highlight the efficiency and accuracy of the proposed method.

The fractional mobile/immobile solute transport model has applications in a wide range of phenomena such as ocean acoustic propagation and heat diffusion. The local radial basis functions (RBFs) method have been applied to many physical and engineering problems because of its simplicity in implementation and its superiority in solving different real-world problems easily. In this article, we propose an efficient local RBFs method coupled with Laplace transform (LT) for approximating the solution of fractional mobile/immobile solute transport model in the sense of Caputo derivative. In our method, first, we employ the LT which reduces the problem to an equivalent time-independent problem. The solution of the transformed problem is then approximated via the local RBF method based on multiquadric kernels. Afterward, the desired solution is represented as a contour integral in the left half complex along a smooth curve. The contour integral is then approximated via the midpoint rule. The main advantage of the LT-RBFs method is the avoiding of time discretization technique due which overcomes the time instability issues, second is its local nature which overcomes the ill-conditioning of the differentiation matrices and the sensitivity of the shape parameter, since the local RBFs method only considers the discretization points in each local domain around the collocation point. Due to this, sparse and well-conditioned differentiation matrices are produced, and third is the low computational cost. The convergence and stability of the numerical scheme are discussed. Some test problems are performed in one and two dimensions to validate our numerical scheme. To check the efficiency, accuracy, and efficacy of the scheme the 2D problems are solved in complex domains. The numerical results confirm the stability and efficiency of the method.

We obtain results concerning the existence of smooth curves on the cone over a (possibly singular) plane curve. As an application, these results are used to prove the existence of certain smooth space curves which are the set-theoretic complete intersection of a cone with some other surface.

The paper discusses an integrated methodology to implement an interactive augmented reality 3D modelling environment with natural interaction, empowered by real-time gesture recognition. The methodology is developed from a geometry-sculpting algorithm based on the use of the subdivision surfaces approach to combine the ease and versatility of interactive modelling even of complex shapes, while maintaining high geometric continuity and smoothness. The interaction with the deformable elements of the geometry’s control cage to be divided uses an optimised version of the Grasp Active Feature/Object Active Feature algorithm developed from hand tracking and gesture recognition based on zero-invasive stereo-infrared techniques. Modelling, combined with an augmented reality environment, allows the modification of geometries having real objects as a reference and, in any case, a general spatial awareness during activities. The methodology was implemented and tested using an advanced mixed-reality headset, the Varjo XR-4, with hi-resolution pass-through and a second-generation Ultraleap for accurate and precise hand tracking.

As the field of robotics advances swiftly, industrial automation has become prevalent in the realms of manufacturing and precision measurement. In robot measurement applications, the original path often originates from the discrete output of CAD models or point cloud data of vision systems, and its measurement path is a linear path composed of discrete feature points. Vibrations are generated by robots when passing through corners between adjacent linear segments. In order to reduce vibration, an algorithm for smoothing the robot’s measurement path based on multiple curves is proposed. Based on the proposed robot scanning measurement path generation algorithm, a robot scanning measurement path is generated. The position and attitude of the scanning path are represented as multiple curves using a position and attitude representation method based on multiple curves. The corners of the position curve and attitude curve are smoothed using a 5th-order B-spline curve. Based on the established robot position tolerance and attitude tolerance constraints and geometric continuity, the control points of the B-spline curve are solved, and corresponding position corner smooth B-spline curves and attitude corner smooth B-spline curves are constructed. Based on the geometric continuity, we use B-spline curves to replace the transition parts of adjacent position corner points and adjacent attitude corner points in the scanning path and then achieve the synchronization of robot position and attitude by the common curve parameter method. Finally, the effectiveness of our proposed path smoothing algorithm was verified through robot joint tracking experiments and scanning measurement experiments.

Collective spin excitations form a fundamental class of excitations in magnetic materials. As their energy reaches down to only a few meV, they are present at all temperatures and substantially influence the properties of magnetic systems. To study the spin excitations in solids from first principles, we have developed a computational scheme based on many-body perturbation theory within the full-potential linearized augmented plane-wave (FLAPW) method. The main quantity of interest is the dynamical transverse spin susceptibility or magnetic response function, from which magnetic excitations, including single-particle spin-flip Stoner excitations and collective spin-wave modes as well as their lifetimes, can be obtained. In order to describe spin waves we include appropriate vertex corrections in the form of a multiple-scattering T matrix, which describes the coupling of electrons and holes with different spins. The electron-hole interaction incorporates the screening of the many-body system within the random-phase approximation. To reduce the numerical cost in evaluating the four-point T matrix, we exploit a transformation to maximally localized Wannier functions that takes advantage of the short spatial range of electronic correlation in the partially filled d or f orbitals of magnetic materials. The theory and the implementation are discussed in detail. In particular, we show how the magnetic response function can be evaluated for arbitrary k points. This enables the calculation of smooth dispersion curves, allowing one to study fine details in the k dependence of the spin-wave spectra. We also demonstrate how spatial and time-reversal symmetry can be exploited to accelerate substantially the computation of the four-point quantities. As an illustration, we present spin-wave spectra and dispersions for the elementary ferromagnet bcc Fe, B2-type tetragonal FeCo, and CrO₂ calculated with our scheme. The results are in good agreement with available experimental data.

Let X be a smooth projective curve of positive genus defined over a number field K. Assume given a Galois covering map x : X → PK and a place v of K. We introduce a local canonical height on X(Kv) associated to x as an integral with logarithmic integrand, generalizing Tate’s local Neron function on an elliptic curve. The resulting global height can be viewed as a ‘Mahler measure’ associated to x. We prove that the local canonical height can be obtained by averaging, and taking a limit, over divisors of higher order Weierstrass points on X. This generalizes previous results by Everest-ni Fhlathuin and Szpiro-Tucker. Our construction of the local canonical height is an application of potential theory on Berkovich curves in the presence of a canonical measure.

Canonical systems and their limits on stable curves

Generic singularities of rays emanating from a space curve in R 3 in all directions with the rate determined by an indicatrix (independent of the point in R 3 ) defined by a surface are classified. Similarly rays emanating from surface defined by an indicatrix given by a curve are also considered. Some applications to control theory are indicated. In this paper we solve two problems on the classification of local geometrical singularities that are related to control theory. We use some techniques from the singularity theory of caustics and wave fronts to study singularities of exponential mappings in a class of control problems which correspond to special integrable Hamiltonian systems with straight lines as extremals. The first problem concerns a control system on a three-dimensional ane space with points q 2 R 3 . We identify the tangent space TqR 3 with R 3 itself. At each point q we choose an indicatrix Iq of admissible velocities _ q = @q @ of motion which we assume is independent of the point q itself. Assume that this set is parametrised locally by a regular mapping (x;y)7! r2(x;y) whose image is a surface M. We shall now write M in place of Iq. An admissible motion is a smooth curve ( )2 R 3 , parametrised by a segment of the (ane) time axis , such that the velocity at each point _ belongs to the set of admissible velocities M. Let qb( ) be the trajectory of an admissible motion of an initial point b2 N, issuing at = 0 from an initial set N, where N is a space curve which is a submanifold in R 3 .

We consider a smooth curve with singular points in the Euclidean space. As a smooth curve with singular points, we have introduced a framed curve or a framed immersion. A framed immersion is a smooth curve with a moving frame and the pair is an immersion. We define an evolute and a focal surface of a framed immersion in the Euclidean space. The evolutes and focal surfaces of framed immersions are generalizations of each object of regular space curves. We give relationships between singularities of the evolutes and of the focal surfaces. Moreover, we consider properties of the evolutes, focal surfaces and repeated evolutes.

Dedicated methods for dealing with curve interpolation and curve smoothing have been developed. One such method, Hurwitz-Radon Matrices (MHR), can be used in reconstruction and interpolation of curves in the plane. The method is based on a family of Hurwitz-Radon (HR) matrices. The matrices are skew-symmetric and possess columns composed of orthogonal vectors. The of Hurwitz-Radon Operator (OHR), built from these matrices, is described. It is shown how to create the orthogonal and discrete OHR and how to use it in a process of curve interpolation and modelling. The method needs suitable choice of nodes, i.e. points of the curve to be reconstructed: nodes should be settled at each local extremum and nodes should be monotonic in one of coordinates (for example equidistance). Application of MHR gives a very good interpolation accuracy in the process of modeling and reconstruction of the curve. Created from the family of N-1 HR matrices and completed with the identity matrix, the system of matrices is orthogonal only for vector spaces of dimensions N = 2, 4 or 8. Orthogonality of columns and rows is very important and significant for stability and high precision of calculations. The MHR method models the curve point by point without using any formula or function. Main features of the MHR method are: the accuracy of curve reconstruction depends on the number of nodes and method of choosing nodes, interpolation of L points of the curve has a computational cost of rank O(L), and the smoothing of the curve depends on the number of OHR operators used to build the average matrix operator. The problem of curve length estimation is also considered. Algorithms and numerical results are presented.

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.