Leonardo Fernández-Jambrina

Universidad Politécnica de Madrid | UPM · E.T.S.I. Navales

In this paper, we make use of an inverse formula that relates the blossom of a NURBS curve, surface or Bézier triangle with its parametrisation, with no explicit reference to the control points and weights of the parametrisation. We make use of this inverse formula to raise and lower the degree elevation and reduction.

In this talk we review the problem of constructing a developable surface patch bounded by two rational or NURBS (Non-Uniform Rational B-spline) curves.

El aula invertida es ya una metodología docente madura. Trasladar parte de las actividades del aula a un entorno no presencial parece aún más apropiado en un contexto de confinamiento y distancia social como el que nos ha tocado vivir. El problema reside en el resto de actividades que se desarrollan o desarrollaban en el aula. Con esta comunicación...

The discovery of accelerated expansion of the Universe opened up the possibility of new scenarios for the doom of our space–time, besides eternal expansion and a final contraction. In this paper, we review the chances that may await our universe. In particular, there are new possible singular fates (sudden singularities, big rip, etc.), but there a...

Pipes and offsets are the sets obtained by displacing the points of their progenitor $ S $ (i.e., spine curve or base surface, respectively) a constant distance $ d $ along normal lines. We review existing results and elucidate the relationship between the smoothness of pipes/offsets and the reach $ R $ of the progenitor, a fundamental concept in F...

Due to the accelerated expansion of the universe, the possibilities for the formation of singularities has changed from the classical Big Bang and Big Crunch singularities to include a number of new scenarios. In recent papers it has been shown that such singularities may appear in inflationary cosmological models with a fractional power scalar fie...

Integral theorems such as Stokes' and Gauss' are fundamental in many parts of physics. For instance, Faraday's law allows computing the induced electric current on a closed circuit in terms of the variation of the flux of a magnetic field across the surface spanned by the circuit. The key point for applying Stokes' theorem is that this surface must...

In Farin (2006) Farin proposed a method for designing Bézier curves with monotonic curvature and torsion. Such curves are relevant in design due to their aesthetic shape. The method relies on applying a matrix M to the first edge of the control polygon of the curve in order to obtain by iteration the remaining edges. With this method, sufficient co...

In this paper we provide a characterisation of rational developable surfaces in terms of the blossoms of the bounding curves and three rational functions Λ, M , ν. Properties of developable surfaces are revised in this framework. In particular, a closed algebraic formula for the edge of regression of the surface is obtained in terms of the function...

In this paper we construct developable surface patches which are bounded by two rational or NURBS curves, though the resulting patch is not a rational or NURBS surface in general. This is accomplished by reparameterizing one of the boundary curves. The reparameterization function is the solution of an algebraic equation. For the relevant case of cu...

In this paper we address the issue of designing developable surfaces with Bezier patches. We show that developable surfaces with a polynomial edge of regression are the set of developable surfaces which can be constructed with Aumann's algorithm. We also obtain the set of polynomial developable surfaces which can be constructed using general polyno...

In Stoica (Int. J. Theor. Phys. 55, 71–80, 2016) a regularization procedure is suggested for regularizing Big Bang singularities in Friedmann-Lemaître-Robertson-Walker (FLRW) spacetimes. We argue that this procedure is only appliable to one case of Big Bang singularities and does not affect other types of singularities.

In Haro, Amorós, and Pan [Phys. Rev. D 93, 084018 (2016)] a new cosmological model is proposed with no big bang singularity in the past, though past geodesically incomplete. This model starts with an inflationary era, follows with a stiff matter dominated period and evolves to accelerated expansion in an asymptotically de Sitter regime in a realist...

In this paper we classify and derive closed formulas for geometric elements of quadrics in rational B\'ezier triangular form (such as the center, the conic at infinity, the vertex and the axis of paraboloids and the principal planes), using just the control vertices and the weights for the quadric patch. The results are extended also to quadric ten...

In [1] a new cosmological model is proposed with no big bang singularity in the past. This model starts with an inflationary era, follows with a stiff matter dominated period and evolves to accelerated expansion in an asymptotically de Sitter regime. We argue that the initial singularity is in fact no big bang but a directional singularity which ca...

In this paper we review the derivation of implicit equations for non-degenerate quadric patches in rational Bezier triangular form. These are the case of Steiner surfaces of degree two. We derive the bilinear forms for such quadrics in a coordinate-free fashion in terms of their control net and their list of weights in a suitable form. Our construc...

In this paper we address the problem of interpolating a spline developable patch bounded by a given spline curve and the first and the last rulings of the developable surface. In order to complete the boundary of the patch a second spline curve is to be given. Up to now this interpolation problem could be solved, but without the possibility of choo...

The present accelerated expansion of the universe has enriched the list of possible scenarios for its fate, singular or not. In this paper a unifying framework for analyzing such behaviors is proposed, based on generalized power and asymptotic expansions of the barotropic index $w$, or equivalently of the deceleration parameter $q$, in terms of the...

In this talk we would like to analyse the appearance of singularities in FLRW cosmological models which evolve close to w = -1, where w is the barotropic index of the universe. We relate small terms in cosmological time around w = -1 with the correspondent scale factor of the universe and check for the formation of singularities.

In this paper, we address the calculation of geometric characteristics of conic sections (axes, asymptotes, centres, eccentricity, foci) given in Bézier form in terms of their control polygons and weights, making use of real and complex projective and affine geometry and avoiding the use of coordinates.

Recently a new type of cosmological singularity has been postulated for infinite barotropic index ω in the equation of state p = ωρ of the cosmological fluid, but vanishing pressure and density at the singular event. Apparently the barotropic index ω would be the only physical quantity to blow up at the singularity. In this talk we would like to di...

In this comment we explain the discrepancies mentioned by the authors between their results and ours about the influence of the gravitational quadrupole moment in the perturbative calculation of corrections to the precession of the periastron of quasielliptical Keplerian equatorial orbits around a point mass. The discrepancy appears to be consequen...

In this paper we characterize barotropic index singularities of homogeneous isotropic cosmological models [M. P. Dabrowski and T. Denkiewicz, Phys. Rev. D 79, 063521 (2009).]. They are shown to appear in cosmologies for which the scale factor is analytical with a Taylor series in which the linear and quadratic terms are absent. Though the barotropi...

In this talk we show a construction for characterising developable surfaces in the form of Bézier triangular patches. It is shown that constructions used for rectangular patches are not useful, since they provide degenerate triangular patches. Explicit constructions of non-degenerate developable triangular patches are provided.

In this talk we review the appearance of new types of singularities (big rip, sudden singularities...) in FLRW cosmological models that have arisen on considering explanations for accelerated expansion of our universe. Comment: 3 pages, ws-procs975x65.cls to appear in Proceedings of 12th Marcel Grossmann Meeting, Paris

We consider FLRW cosmological models with standard Friedmann equations, but leaving free the equation of state. We assume that the dark energy content of the universe is encoded in an equation of state $p=f(\rho)$, which is expressed with most generality in the form of a power expansion. The inclusion of this expansion in Friedmann equations allows...

In this lecture we will show some properties of a singularity-free solution to Einstein's equations and its accordance with some theorems dealing with singularities. We will also discuss the implications of the results. Comment: 5 pp. Published in Proceedings of ERE'91

In this lecture we deal with the construction of surface densities for the angular momentum of the sources of asymptotically flat vacuum stationary axisymmetric spacetimes. These sources arise from the discontinuities of the twist potential. The result will be applied to the Kerr metric to obtain an integrable density which can be viewed as the reg...

In this talk it is shown a way for constructing magnetic surface sources for stationary axisymmetric electrovac spacetimes possessing a non-smooth electromagnetic Ernst potential. The magnetic moment density is related to this lack of smoothness and its calculation involves solving a linear elliptic differential equation. As an application the resu...

In this lecture a new formalism for constructing electromagnetic surface sources for static axisymmetric electrovacs is presented. The electrostatic and magnetostatic sources are derived from the discontinuities of the scalar potentials. This formalism allows the inclusion of two kinds of dipole sources: Sheets of dipoles and the dipole moment of a...

In this talk relativistic corrections due to Geroch-Hansen multipoles for perihelion precession and node line precession of orbits in a stationary axially symmetric vacuum spacetime endowed with a plane of symmetry will be shown. Patterns of regularity will be discussed. Comment: 5 pp, sprocl.sty, Proceedings of ERE'00

In this talk we show a stiff fluid solution of the Einstein equations for a cylindrically symmetric spacetime. The main features of this metric are that it is non-separable in comoving coordinates for the congruence of the worldlineS of the fluid and that it yields regular curvature invariants. Comment: 4 pp. Proceedings of ERE'96

In this talk we shall show a perfect fluid cosmological model and its properties. The model possesses an orthogonally transitive abelian two-dimensional group of isometries that corresponds to cylindrical symmetry. The matter content is a stiff fluid that satisfies the energy and generic conditions. The metric is not separable in comoving coordinat...

In this talk we would like to review recent results on non-singular cosmological models. It has been recently shown that among stiff perfect fluid inhomogeneous spacetimes the absence of singularities is more common than it was expected in the literature. We would like to generalize these results and apply them to other matter sources.

In this talk a previous theorem on geodesic completeness of diagonal cylindrical spacetimes will be generalized to cope with the nondiagonal case. A sufficient condition for such spacetimes to be causally geodesically complete will be given

In this talk a sufficient condition for a diagonal orthogonally transitive cylindrical $G_2$ metric to be geodesically complete is given. The condition is weak enough to comprise all known diagonal perfect fluid cosmological models that are non-singular.

A sufficient condition for an orthogonally transitive G2 cylindrical spacetime to be singularity-free is shown. The condition is general enough to comprise all known geodesically complete perfect fluid cosmologies.

The possibility of obtaining an open set of regular cosmological models is discussed. Cylindrical stiff perfect fluid cosmologies are studied in detail. The condition for geodesic completeness is easy to check. A large family of non-singular models is found therein.

In this talk the possibility of constructing geodesically complete inhomogeneous stiff fluid cosmologies is discussed. A family with infinite parameters is derived. A wide and easy to implement sufficient condition for geodesic completeness is shown.

In this talk we extend a family of geodesically complete $G_{2}$ stiff fluid cosmological models to the case in which the velocity of the fluid is not orthogonal to the gradient of the transitivity surface element.

In this paper we provide a thorough classification of Friedman-Lema\^itre-Robertson-Walker (FLRW) cosmological models in terms of the strong or weak character of their singularities according to the usual definitions. The classification refers to a generalised Puiseux power expansion of the scale factor of the model around a singular event.

In this talk we analyze the effect of recently proposed classes of sudden future singularities on causal geodesics of FLRW spacetimes. Geodesics are shown to be extendible and just the equations for geodesic deviation are singular, although tidal forces are not strong enough to produce a Big Rip. Comment: 4 pp. Published in Spanish Relativing Meeti...

We consider perturbative modifications of the Friedmann equations in terms of energy density corresponding to modified theories of gravity proposed as an alternative route to comply with the observed accelerated expansion of the universe. Assuming that the present matter content of the universe is a pressureless fluid, the possible singularities th...

In this Letter we study the final fate of the universe in modified theories of gravity. As compared with general relativistic formulations, in these scenarios the Friedmann equation has additional terms which are relevant for low density epochs. We analyze the sort of future singularities to be found under the usual assumption the expanding Univers...

En el marco de la reforma de las titulaciones con motivo del Espacio Europeo de Educación un grupo de profesores hemos coordinado, durante el curso 2008-2009, todas las asignaturas básicas de primer curso y una más de segundo curso en la Escuela Técnica Superior de Ingenieros Navales. Las actividades realizadas son: a) Coordinación de todas las asi...

It has been recently proved that rational quadratic circles in standard Bézier form are parameterized by chord-length. If we consider that standard circles coincide with the isoparametric curves in a system of bipolar coordinates, this property comes as a straightforward consequence. General curves with chord-length parametrization are simply the a...

This chapter is devoted to the origins of relativistic astrophysics, both from the theoretical and observational point of view. Supernova explosions, pulsars, active galactic nuclei and gamma-ray bursts are some of the observed processes that are the object of this discipline. On the other hand, the intriguing features of black holes, singularities...

FLRW models filled with just dark energy are shown to have a finite past, since causal geodesics cannot be extended beyond a certain proper time. It is shown that curvature measured along causal geodesics becomes infinity on travelling to the past, though curvature scalars tend to zero. Furthermore the time measured by free-falling observers from c...

In this Letter we analyse the possibility of having homogeneous isotropic cosmological models with observers reaching t=∞ in finite proper time. It is shown that just observationally-suggested dark energy models with w∈(−5/3,−1) show this feature and that they are endowed with an exotic curvature singularity. Furthermore, it is shown that non-accel...

In this paper a linear algorithm is derived for constructing B-spline control nets for spline developable surfaces of arbitrary degree and number of pieces. Control vertices are written in terms of five free parameters related to the type of developable surface. Aumann's algorithm for constructing Bézier developable surfaces is recovered as a parti...

In this paper causal geodesic completeness of Friedmann-Lema\^\itre-Robertson-Walker (FLRW) cosmological models is analyzed in terms of generalized power expansions of the scale factor in coordinate time. The strength of the found singularities is discussed following the usual definitions due to Tipler and Kr\'olak. It is shown that while classical...

A conjecture stated by Raychaudhuri which claims that the only physical perfect fluid non-rotating non-singular cosmological models are comprised in the Ruiz-Senovilla and Fernandez-Jambrina families is shown to be incorrect. An explicit counterexample is provided and the failure of the argument leading to the result is explicitly pointed out. Comm...

In this paper we analyze the effect of recently proposed classes of sudden future singularities on causal geodesics of FLRW spacetimes. Geodesics are shown to be extendible and just the equations for geodesic deviation are singular, although tidal forces are not strong enough to produce a Big Rip. For the sake of completeness, we compare with the t...

In this paper we analyze Abelian diagonal orthogonally transitive space-times with spacelike orbits for which the matter content is a stiff perfect fluid. The Einstein equations are cast in a suitable form for determining their geodesic completeness. A sufficient condition on the metric of these space-times is obtained, that is fairly easy to check...

A cylindrically symmetric perfect fluid spacetime with no curvature singularity is shown. The equation of state for the perfect fluid is that of a stiff fluid. The metric is diagonal and non-separable in comoving coordinates for the fluid. It is proven that the spacetime is geodesically complete and globally hyperbolic. Comment: LaTeX 2e, 8 pages

In this paper a new method is derived for constructing electromagnetic surface sources for stationary axisymmetric electrovac spacetimes endowed with non-smooth or even discontinuous Ernst potentials. This can be viewed as a generalization of some classical potential theory results, since lack of continuity of the potential is related to dipole den...

83C05 Einstein's equations (general structure, canonical formalism, Cauchy problems) 83D05 Relativistic gravitational theories other than Einstein's, including asymmetric field theories

In this paper we analyse the possibility of constructing singularity-free inhomogeneous cosmological models with a pure radiation field as matter content. It is shown that the conditions for regularity are very easy to implement and therefore there is a huge number of such spacetimes.

In this paper a family of non-singular cylindrical perfect fluid cosmologies is derived. The equation of state corresponds to a stiff fluid. The family depends on two independent functions under very simple conditions. A sufficient condition for geodesic completeness is provided. Comment: 7 pages, RevTeX4

The present volume contains the expanded lectures of a meeting on relativistic astrophysics, the goal of which was to provide a modern introduction to specific aspects of the field for young researchers, as well as for nonspecialists from related areas. Particular emphasis is placed on the theory of black holes and evolution, relativistic stars and...

This chapter is devoted to the origins of relativistic astrophysics, both from the theoretical and observational point of view. Supernova explosions, pulsars, active galactic nuclei and gamma-ray bursts are some of the observed processes that are the object of this discipline. On the other hand, the intriguing features of black holes, singularities...

A theorem stated by Raychaudhuri which claims that the only physical non-singular cosmological models are comprised in the Ruiz-Senovilla family is shown to be incorrect. An explicit counterexample is provided and the failure of the argument leading to the theorem is explicitly pointed out.

En esta memoria dedicada a la interpretación de soluciones de las ecuaciones de Einstein se han abordado las siguientes cuestiones: - construcción de un formalismo exterior para la descripción de campos electromagnéticos con simetrías estacionaria y axial dentro de la teoría de la relatividad general. - un nuevo enfoque para abordar la generalizaci...

The first terms of the general solution for an asymptotically flat stationary axisymmetric vacuum spacetime endowed with an equatorial symmetry plane are calculated from the corresponding Ernst potential up to seventh order in the radial pseudospherical coordinate. The metric is used to determine the influence of high order multipoles in the perihe...

In this paper a theorem is derived in order to provide a wide sufficient condition for an orthogonally transitive cylindrical spacetime to be singularity-free. The applicability of the theorem is tested on examples provided by the literature that are known to have regular curvature invariants.

In this paper a new formalism based on exterior differential systems is derived for perfect-fluid spacetimes endowed with an abelian orthogonally transitive G2 group of motions acting on spacelike surfaces. This formulation allows simplifications of Einstein equations and it can be applied for different purposes. As an example a singularity-free me...

83C05 Einstein's equations (general structure, canonical formalism, Cauchy problems) 83C15 Exact solutions 83C22 Einstein-Maxwell equations 83C50 Electromagnetic fields

A method for interpreting discontinuities of the twist potential of vacuum stationary axisymmetric solutions of Einstein's equations is introduced. Surface densities for the angular momentum of the source can be constructed after solving a linear partial differential equation with boundary conditions at infinity. This formalism is applied to the Ke...

We show that the solution published in the paper by Senovilla [Phys. Rev. Lett. 64, 2219 (1990)] is geodesically complete and singularity-free. We also prove that the solution satisfies the stronger energy and causality conditions, such as global hyperbolicity, the strong energy condition, causal symmetry, and causal stability. A detailed discussio...

In this talk we show an example of a thorough e-course on Computer-Aided Geometric Design (CAGD), which makes use of Java animations. The course describes basic algorithms for the layout of Bézier, rational, B-spline curves and surfaces, with special emphasis on ruled, translational, developable, Coons and revolution surfaces.

En el marco de la reforma de las titulaciones con motivo de la puesta en marcha del Espacio Europeo de Educación Superior, un grupo de profesores hemos decidido coordinar todas las asignaturas básicas de primer curso y una asignatura de segundo curso en la Escuela Técnica Superior de Ingenieros Navales de la Universidad Politécnica de Madrid con el...

Developable surfaces are defined as zero gaussian curvature surfaces (intrinsically flat). That is, plane patches that are curved by just folding, rolling or cutting, but without stretching or combing. Useful for depicting steel plates in naval industry, cloth in textile industry. . . But they are difficult to include in the NURBS formulation for the...

En esta comunicación se explicará la evolución temporal de los resultados del aprendizaje de los alumnos de una asignatura de Matemáticas de segundo curso de Ingeniería Naval (Métodos Matemáticos de la Ingeniería I) desde un sistema de clase magistral con exámenes parciales y finales (curso 2003-4) hasta un sistema basado en la evaluación continua...