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Funding for open access charge: Universidad de Granada / CBUA., Standard Offset surfaces are defined as locus of the points which are at constant distance along the unit normal direction from the generator surfaces. Offset are widely used in various practical applications, such as tolerance analysis, geometric optics and robot path-planning. In some of the engineering applications, we need to extend the concept of standard offset to the generalized offset where distance offset is not necessarily constant and offset direction are not necessarily along the normal direction. Normally, a generalized offset is functionally more complex than its progenitor because of the square root appears in the expression of the unit normal vector. For this, an approximation method of its construction is necessary. In many situation it is necessary to fill or reconstruct certain function defined in a domain in which there is a lack of information inside one or several sub-domains (holes). In some practical cases, we may have some specific geometrical constrains, of industrial or design type, for example, the case of a specified volume inside each one of these holes. The problem of filling holes or completing a 3D surface arises in all sorts of computational graphics areas, like CAGD, CAD-CAM, Earth Sciences, computer vision in robotics, image reconstruction from satellite and radar information, etc. In this work we present an approximation method of filling holes of the generalized offset of a surface when there is a lack information in a sub-domain of the function that define it. We prove the existence and uniqueness of solution of this problem, we show how to compute it and we establish a convergence result of this approximation method. Finally, we give some graphical and numerical examples., Universidad de Granada / CBUA

We present a smoothing method of a set of points to be approximated from a given boundary value problem for the modified Helmholtz equation. A solution of such problem will be called MH-spline. Two convergence results are carefully established. Some numerical and graphic examples are shown in order to test the validity of the method.

This article deals with a numerical approximation method using an evolutionary partial differential equation (PDE) by discrete variational splines in a finite element space. To formulate the problem, we need an evolutionary PDE equation with respect to the time and the position, certain boundary conditions and a set of approximating points. We show the existence and uniqueness of the solution and we study a computational method to compute such a solution. Moreover, we established a convergence result with respect to the time and the position. We provided several numerical and graphic examples of approximation in order to show the validity and effectiveness of the presented method.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 34: 5–18, 2018

This paper deals with an approximation problem concerning vector fields through the new notion of div–rot variational splines. The minimizing problem is addressed in a finite element space through the choice of some semi-norms based on decomposition of the divergence operator and vector fields into a form with a rotational part. We study the existence and the uniqueness of the solution of such a problem. Then, a convergence result and an estimation of the error are established. Some numerical and graphical examples are analyzed in order to prove the validity of our method. Furthermore, we compare and show how our method improves upon one existing in the literature.

This paper deals with the construction and characterization of discrete variational PDE splines. To formulate the problem, we need an evolutionary PDE equation, certain boundary conditions and a set of points to approximate. We show the existence and the uniqueness of the solution of such a problem and we establish a convergence result of a discrete variational evolutionary PDE spline to a given function. We present several numerical and graphical examples of construction and approximation of surfaces in order to prove the validity of the presented method.

Smoothing of noisy data has always been a topic of interest in many areas where computer simulations have been performed, including natural sciences as well as economics and social sciences. In this paper we present an approximation method of explicit curves or surfaces from exact and noisy data by fairness cubic or bicubic splines . A variational problem of explicit curves or surfaces is obtained by minimizing a quadratic functional in a space of cubic or bicubic splines from noisy data. We show the existence and uniqueness of this problem as long as a convergence result especially for noisy data is carefully established. We analyze some numerical and graphical examples using fictional noisy data in order to prove the validity of our method.

In this paper, we present an interpolation method for curves from a data set by means of the optimization of the parameters of a quadratic functional in a space of parametric cubic spline functions. The existence and the uniqueness of this problem are shown. Moreover, a convergence result of the method is established in order to justify the method presented. The aforementioned functional involves some real non-negative parameters; the optimal parametric curve is obtained by the suitable optimization of these parameters. Finally, we analyze some numerical and graphical examples in order to show the efficiency of our method.

This paper addresses the problem of constructing some free-form curves and surfaces from given to different types of data: exact and noisy data. We extend the theory of Dm-splines over a bounded domain for noisy data to the smoothing variational vector splines. Both results of convergence for respectively the exact and noisy data are established, as soon as some estimations of errors are given.

We deal with a smoothing method of constructing some discontinuous curve or surface from a Lagrangean data and tangent conditions. Such method is based on the theory of smoothing variational splines conveniently adapted to introduce the tangent condition and the discontinuity set. To show the efficiency of our method we finish this work by a convergence result and some numerical and graphical examples.

In this paper we present an approximation method of surfaces by a new type of splines, which we call fairness bicubic splines, from a given Lagrangian data set. An approximating problem of surface is obtained by minimizing a quadratic functional in a parametric space of bicubic splines. The existence and uniqueness of this problem are shown as long as a convergence result of the method is established. We analyze some numerical and graphical examples in order to prove the validity of our method.

We present in this paper an approximation method of curves from sets of Lagrangian data and vectorial tangent subspaces. We define a discrete smoothing fairness spline with tangent conditions by minimizing certain quadratic functional on finite element spaces. Convergence theorem is established and some numerical and graphical examples are analyzed in order to show the validity and the effectiveness of this paper.

In this paper we present a numerical approximation of curves and surfaces from a given scattered data set. An approximating curve or surface problem is obtained by minimizing a quadratic functional in a parametric finite element space, its solution is called a discrete smoothing variational spline. The existence and uniqueness of this problem are shown, as long as the convergence of the method is established. Finally some particular examples are given.

Resumen tomado de la publicación A comienzos del curso académico 2009-2010 se impartió un curso cero para los alumnos de nuevo ingreso en la Escuela de Arquitectura de la Universidad de Granada como proyecto piloto para consolidar, de cara a la llegada del nuevo Plan de Estudios, el siguiente curso. Se observa que los alumnos que cada año llegan a primero tienen una base de conocimientos muy dispar en matemáticas debido, entre otros factores, a que los conceptos que han aprendido en Bachillerato son distintos en función del centro en el que hayan estudiado. Por ello se organizan sesiones para repasar los conceptos matemáticos básicos que los alumnos deben conocer para estudiar, no solamente las asignaturas de matemáticas, sino también otras como física, estructuras o geometría. Se pretende homogeneizar los conocimientos matemáticos de todos los alumnos. Asimismo, se organizan sesiones para familiarizar al alumno nuevo con la universidad y todos sus servicios, en especial los que son de interés para ellos. Al finalizar se realizan a los alumnos algunas cuestiones relacionadas con la organización y contenidos del curso. Los resultados son muy satisfactorios. A la pregunta ¿te has arrepentido de apuntarte al curso cero?, todos los alumnos contestaron que no. Navarra Universidad Pública de Navarra. Biblioteca Universitaria; Campus de Arrosadia; 31006 Pamplona; Tel. +34948169060; Fax +34948169069; bupna@enavarra.es ESP

This is a note on the paper [A. Kouibia, A.J. López-Linares, M. Pasadas, Approximation of discontinuous curves and surfaces with tangent conditions, J. Comput. Appl. Math. 193 (2006) 51–64]. We consider the constructing problem of a discontinuous parametric curve or surface from a finite set of points and tangent conditions. We develop a method based on the theory of discrete smoothing variational splines conveniently adapted to introduce the tangent conditions and the discontinuity set. We give a convergence result and we analyze some numerical and graphical examples in order to illustrate the effectiveness of the presented method.

We present in this paper a discrete problem of constructing some parametric surfaces with parallelism conditions from some given Lagrangean data. We consider the problem of fitting some scattered points with a surface pseudo-parallel to a given reference surface in a finite element space. Some convergence results are shown. Finally, we analyze some graphical examples in order to prove the validity and the effectiveness of this method.

In this paper we present an approximation problem of parametric curves and surfaces from a Lagrange or Hermite data set. In particular, we study an interpolation problem by minimizing some functional on a Sobolev space that produces the new notion of interpolating variational spline. We carefully establish a convergence result. Some specific cases illustrate the generality of this work.

This paper deals with the problem of constructing some free-form curves and surfaces from given Lagrangian and/or Hermite data. We define the smoothing variational spline function by minimizing a certain quadratic functional in a Sobolev space and establish the convergence of the associated method.