J. Monterde

University of Valencia | UV · Matemàtiques

The relationship between Salkowski curves, a family of slant helices with constant curvature and non-constant torsion, and the family of spherical epicycloid curves is studied. It is shown that, for some values of the parameter defining the Salkowski curve, the curve is the image by a shear transformation along the z-axis of a spherical epicycle. T...

Holditch’s theorem is a classical geometrical result on the areas of a given closed curve and another one, its Holditch curve, which is constructed as the locus of a fixed point dividing a chord of constant length that moves with its endpoints over the given curve and that returns back to its original position after some full revolution. Holditch c...

A well known result states that the curvature of the normal bundle of an n -manifold immersed in the Euclidean space, $${\mathbb {R}}^{n+k}$$ R n + k , vanishes if and only if the curvature locus at any point is a convex polytope. In the case of 3-manifolds, the curvature locus is, generically, a triangle. In this paper we determine, among all the...

An introduction to non-Euclidean geometry is given and a new approach to prove a generalization of Holditch's theorem in 2-dimensional constant curvature manifolds is presented. Moreover, the same procedure also makes possible to obtain generalized versions of Barbier's theorem for constant width curves and of Steiner's formulae for parallel curves...

After a review of two methods of construction of Bertrand curves, we characterize Bertrand curves defined from Salkowski curves as the only family of Bertrand curves which are slant helices.

An immersed surface in \({{\mathbb {R}}}^4\) is said to has constant Jordan angles (CJA) if the angles between its tangent planes and a fixed plane do not depend on the choice of the point. The constant Jordan angles surfaces in \({{\mathbb {R}}}^4\) has been proved to exist, Bayard et al. (Geom Dedicata 162:153–176, 2013), but there are only expli...

We first define a complex angle between two oriented spacelike planes in 4-dimensional Minkowski space, and then study the constant angle surfaces in that space, i.e. the oriented spacelike surfaces whose tangent planes form a constant complex angle with respect to a fixed spacelike plane. This notion is the natural Lorentzian analogue of the notio...

Holditch’s theorem is an old result on the area generated by a moving chord for closed planar curves. Some generalizations of this result have been given before, but none of these follows the same natural construction of the plane but done in the space. In this work, the notion of Holditch surface is defined, some properties of these surfaces are p...

We propose an invariant three-point curvature approximation for plane curves based on the arc of a parabolic sector, and we analyze how closely this approximation is to the true curvature of the curve. We compare our results with the obtained with other invariant three-point curvature approximations. Finally, an application is discussed.

In plane geometry, Holditch's theorem states that if a chord of fixed length is allowed to rotate inside a convex closed curve, then the locus of a point on the chord a distance p from one end and a distance q from the other is a closed curve whose area is less than that of the original curve by πpq. In this article we obtain, first, sufficient con...

Recently we have shown that the distance trisector curve is a transcendental curve. Since from the computational point of view this implies that no closed expression to describe the curve in algebraic terms can be found, it is still of interest to know how to approximate it efficiently by means of polynomial or rational functions. We discuss here s...

The PDE under study here is a general fourth-order linear elliptic Partial Differential Equation. Having prescribed the boundary control points, we provide the explicit expression of the whole control net of the associated PDE Bézier surface. In other words, we obtain the explicit expressions of the interior control points as linear combinations of...

We obtain a characterization for spherical immersions in R4 in terms of local invariants. Besides the already known fact that the spherical immersions have to be semiumbilical, another condition among the Gauss curvature, the mean curvature and the resultant has to be satisfied.

We present a survey of some results and questions related to the notion of scalar curvature in the setting of symplectic supermanifolds.

Computation of the blending surface of two given spheres is discussed in this paper. The blending surface (or skin), although not uniquely defined in the literature, is normally required to touch the given spheres in plane curves (i.e., in circles). The main advantage of the presented method over the existing ones is the minimization of unwanted di...

A method is given for generating harmonic tensor product Bézier surfaces and the explicit expression of each point in the control net is provided as a linear combination of prescribed boundary control points. The matrix of scalar coefficients of these combinations works like a mould for harmonic surfaces. Thus, real-time manipulation of the resulti...

We discuss several partial solutions to the so-called “coquecigrue problem” of Loday; these solutions parallel, but also generalize in several directions, the classical Lie group-Lie algebra correspondence. Our study highlights some clear similarities between the split and nonsplit cases and leads us to a general unifying scheme that provides an an...

Given a supervector bundle $E = E_0\oplus E_1 \to M$, we exhibit a parametrization of Quillen superconnections on $E$ by graded connections on the Cartan-Koszul supermanifold $(M;\Omega (M))$. The relation between the curvatures of both kind of connections, and their associated Chern classes, is discussed in detail. In particular, we find that Cher...

We show that the distance trisector curve is not an algebraic curve, as was conjectured in the founding paper by T. Asano, J. Matousek and T. Tokoyama: "The distance trisector curve", Advances in Math., 212, 338-360 (2007).

An isoperimetric type problem for primitive Pythagorean hodograph curves is studied. We show how to compute, for each possible degree, the Pythagorean hodograph curve of a given perimeter enclosing the greatest area. We also discuss the existence and construction of smooth solutions, obtaining a relationship with an interesting sequence of Appell p...

We give a method using the complex functions associated to the Gauss map to construct examples of non-hyperspherical immersions of ℝ2 in R 4 with two globally defined orthogonal families of asymptotic lines having an isolated inflection point.

We show a possible implementation of the RSA algorithm with Maxima CAS. Theoretical back-ground and numerous examples are given, along with the code. We also implement a simplified version of the digital signature method and comment on some ideas for using this material in the classroom.

We present an explicit polynomial solution method for surface generation. In this case the surface in question is characterized by some boundary configuration whereby the resulting surface conforms to a fourth order linear elliptic Partial Differential Equation, the Euler–Lagrange equation of a quadratic functional defined by a norm. In particular,...

We approach surface design by solving a linear third order Partial Differential Equation (PDE). We present an explicit polynomial solution method for triangular Bézier PDE surface generation characterized by a boundary configuration. The third order PDE comes from a symmetric operator defined here to overcome the anisotropy drawback of any operator...

We approach surface design by solving second-order and fourth-order Partial Differential Equations (PDEs). We present many methods for designing triangular Bézier PDE surfaces given different sets of prescribed control points and including the special cases of harmonic and biharmonic surfaces. Moreover, we introduce and study a second-order and a f...

A new method to build a ring torus is given using only pieces obtained by Villarceau sections. These pieces have the interesting characteristic that all of them are congruent, so we can build a ring torus using an even number of copies of a unique piece. A template of the moon-shaped piece is given.

Given a supermanifold (M,A) which carries a supersymplectic form ω, we study the Fedosov structures that can be defined on it, through a set of tensor fields associated to any symplectic connection ∇. We give explicit recursive expressions for the resulting curvature and study the particular case of a base manifold M with constant holomorphic secti...

In the paper [Salkowski, E., 1909. Zur Transformation von Raumkurven, Mathematische Annalen 66 (4), 517–557] published one century ago, a family of curves with constant curvature but non-constant torsion was defined. We characterize them as space curves with constant curvature and whose normal vector makes a constant angle with a fixed line. The re...

Two methods to generate tensor-product Bézier surface patches from their boundary curves and with tangent conditions along them are presented. The first one is based on the tetraharmonic equation: we show the existence and uniqueness of the solution of with prescribed boundary and adjacent to the boundary control points of a n×n Bézier surface. The...

We give a full characterization of helical polynomial curves of any degree and a simple way to construct them. Existing results about Hermite interpolation are revisited. A simple method to select the best quintic interpolant among all possible solutions is suggested.

No Given a prescribed boundary of a Bezier surface we compare the Bezier surfaces generated by two different methods, i.e. the Bezier surface minimising the Biharmonic functional and the unique Bezier surface solution of the Biharmonic equation with prescribed boundary. Although often the two types of surfaces look visually the same, we show that t...

Given a prescribed boundary of a Bézier surface, we compare the Bézier surfaces generated by two different methods, i.e., the Bézier surface minimising the biharmonic functional and the unique Bézier surface solution of the biharmonic equation with prescribed boundary. Although often the two types of surfaces look visually the same, we show that th...

We give a method to generate polynomial approximations to constant mean curvature surfaces with prescribed boundary. We address this problem by finding triangular Bézier extremals of the CMC-functional among all polynomial surfaces with a prescribed boundary. Moreover, we analyze the C1\mathcal{C}^1 problem, we give a procedure to obtain solutions...

The only surface whose level curves of the Gauss curvature are nongeodesic biharmonic curves and such that the gradient lines are geodesics is, up to local isometries, the revolution surface defined by Caddeo-Montaldo-Piu. Riassunto: L'unica superficie di cui le curve livellate della curvatura di Gauss sono curve biarmoniche nongeodetiche e tale ch...

An intrinsic description of the Hamilton–Cartan formalism for first-order Berezinian variational problems determined by a submersion of supermanifolds is given. This is achieved by studying the associated higher-order graded variational problem through the Poincaré–Cartan form. Noether theorem and examples from superfield theory and supermechanics...

Without Abstract

An intrinsic description of the Hamilton-Cartan formalism for first-order Berezinian variational problems determined by a submersion of supermanifolds is given. This is achieved by studying the associated higher-order graded variational problem through the Poincaré-Cartan form. Noether theorem and examples from superfield theory and supermechanics...

In this paper we present a method for generating Bézier surfaces from the boundary information based on a general 4th-order PDE. This is a generalisation of our previous work on harmonic and biharmonic Bézier surfaces whereby we studied the Bézier solutions for Laplace and the standard biharmonic equation, respectively.Here we study the Bézier solu...

It is shown how to study higher order variational problems in graded fiber bundles through the Poincaré-Cartan form.

A polynomial curve of degree 5, α, is a helix if and only if both ∥α′∥ and ∥α′∧α″∥ are polynomial functions.

Curves in ${\mathbb R}^n$ for which the ratios between two consecutive curvatures are constant are characterized by the fact that their tangent indicatrix is a geodesic in a flat torus. For $n= 3,4$, spherical curves of this kind are also studied and compared with intrinsic helices in the sphere.

We present a new method of surface generation from prescribed boundaries based on the elliptic partial differential operators. In particular, we focus on the study of the so-called harmonic and biharmonic Bézier surfaces. The main result we report here is that any biharmonic Bézier surface is fully determined by the boundary control points. We comp...

We study polynomial solutions in the Bezier form of the wave equation in dimensions one and two. We explicitly determine which control points of the Bezier solution at two difierent times flx the solution. Along this paper the approach will be difierent. We shall try to flnd solutions of the wave equation with the value of the function at time t =...

The Plateau–Bézier problem consists in finding the Bézier surface with minimal area from among all Bézier surfaces with prescribed border. An approximation to the solution of the Plateau–Bézier problem is obtained by replacing the area functional with the Dirichlet functional. Some comparisons between Dirichlet extremals and Bézier surfaces obtaine...

We study some methods of obtaining approximations to surfaces of minimal area with prescribed border using triangular Bézier patches. Some methods deduced from a variational principle are proposed and compared with some masks.

We study the graded Euler–Lagrange equations from the viewpoint of graded Poincaré–Cartan forms. An application to a certain class of solutions of the Batalin–Vilkoviski master equation is also given.

We study the Plateau problem restricted to polynomial sur- faces using techniques coming from the theory of Computer Aided Ge- ometric Design. The results can be used to obtain polynomial approxi- mations to minimal surfaces. The relationship between harmonic Bezier surfaces and minimal surfaces with free boundaries is shown.

Given a symplectic form and a pseudo-Riemannian metric on a manifold, a nondegenerate even Poisson bracket on the algebra of differential forms is defined and its properties are studied. A comparison with the Koszul–Schouten bracket is established.

We provide an intrinsic description of the notion of modular class for an even symplectic manifold and study its properties in this coordinate-free formalism.

There are minimal surfaces admitting a Bézier form. We study the properties that the associated net of control points must satisfy. We show that in the bicubical case all minimal surfaces are, up to an affine transformation, pieces of the Enneper’s surface.

We prove that if an open Non Uniform Rational B-Spline curve of order k has a singular point, then it belongs to both curves of order k — 1 defined in the k — 2 step of the de Boor algorithm. Moreover, both curves are tangent at the singular point.

The Hamilton–Cartan formalism in supermechanics is developed, the graded structure on the manifold of solutions of a variational problem defined by a regular homogeneous Berezinian Lagrangian density is determined and its graded symplectic structure is analyzed. The graded symplectic structure on the manifold of solutions of a classical regular Lag...

We prove that if an nth degree rational Bézier curve has a singular point, then it belongs to the two (n − 1)th degree rational Bézier curves defined in the (n − 1)th step of the de Casteljau algorithm. Moreover, both curves are tangent at the singular point. A procedure to construct Bézier curves with singularities of any order is given.  2001 El...

We define the divergence operators on a graded algebra, and we show that, given an odd Poisson bracket on the algebra, the operator that maps an element to the divergence of the hamiltonian derivation that it defines is a generator of the bracket. This is the "odd laplacian", $\Delta$, of Batalin-Vilkovisky quantization. We then study the generator...

We propose a definition of Jacobi—Nijenhuis structures, that includes the Poisson—Nijenhuis structures as a particular case. The existence of a hierarchy of compatible Jacobi structures on a Jacobi—Nijenhuis manifold is also obtained.RésuméNous proposons une définition des structures de Jacobi—Nijenhuis qui comprennent comme cas particulier les str...

Homogeneous graded metrics over split ℤ2-graded manifolds whose Levi-Civita connection is adapted to a given splitting, in the sense recently introduced by Koszul, are completely described. A subclass of such is singled out by the vanishing of certain components of the graded curvature tensor, a condition that plays a role similar to the closedness...

A proof of the relative version of Frobenius theorem for a graded submersion, which includes a very short proof of the standard graded Frobenius theorem is given. Involutive distributions are then used to characterize split graded manifolds over an orientable base, and split graded manifolds whose Batchelor bundle has a trivial direct summand. Appl...

One-to-one correspondences are established between the set ofall nondegenerate graded Jacobi operators of degree -1 defined onthe graded algebra W(M)\Omega (M) of differential forms on a smooth, oriented,Riemannian manifold M, the space of bundle isomorphisms L:TM ® TML:TM \to TM , and the space of nondegenerate derivations of degree 1 havingnull...

Among all the homogeneous Riemannian graded metrics on the algebra of differential forms, those for which the exterior derivative is a Killing graded vector field are characterized. It is shown that all of them are odd, and are naturally associated to an underlying smooth Riemannian metric. It is also shown that all of them are Ricci-flat in the gr...

We study the graded Poisson structures defined on Ω(M), the graded algebra of differential forms on a smooth manifoldM, such that the exterior derivative is a Poisson derivation. We show that they are the odd Poisson structures previously studied by Koszul, that arise from Poisson structures onM. Analogously, we characterize all the graded symplect...

We express the compatibility conditions that a Poisson bivector and a Nijenhuis tensor must fulfil in order to be a Poisson-Nijenhuis structure by means of a graded Lie bracket. This bracket is a generalization of Schouten and Frlicher-Nijenhuis graded Lie brackets defined on multivector fields and on vector valued differential forms respectively.

In this note we discuss the existence and projectability of graded extensions of ordinary Poisson brackets. We will show that there are topological obstructions to both problems. To prove it we use a new algebraic characterization of graded Poisson brackets on graded manifolds based on a characterization of derivations on the exterior algebra of a...

We state and prove the theorem of existence and uniqueness of solutions to ordinary superdifferential equations on supermanifolds. It is shown that any supervector field, X = X0 + X1, has a unique integral flow, Г: 1¦1 x (M, AM) → (M, AM), satisfying a given initial condition. A necessary and sufficient condition for this integral flow to yield an...

We give a characterization of graded symplectic forms by studying the module of derivations of a graded sheaf. When the graded sheaf is the sheaf of differentiable forms on the underlying manifold M, we find canonical liftings from metrics on TM to odd symplectic forms, and from symplectic forms on M and metrics on TM to even symplectic forms. Thes...

A simple procedure to help calculate the electrostatic potential at any point inside an ionic crystal is proposed and tested. The rationale for the mathematical algorithm to calculate lattice sums is based on Ewald's technique. The method is discussed with regard to the dimensions and shape of the crystal lattice. Electrostatic potential for NaCl a...

The unimolecular decomposition of the most simple amidine (formamidine) was examined by means of ab initio molecular orbital calculations. The Hartree-Fock method in the linear combination of atomic orbitals approximation with the 4-31G basis set was used. A complete potential hypersurface was established and the stationary points representing the...

Los cálculos farmacocinéticos que exige la interpretación de niveles plasmáticos de medicamentos puede convertirse en una de las barreras que impidan su difusión y consecuentemente su aplicación a muchas de las situaciones clínicas en las cuales es obligado, si se desea realizar una terapéutica eficaz. Es decir, obtener el máximo rendimiento terapé...

Given a Poisson–Nijenhuis manifold, a two-parameter family of Poisson– Nijenhuis structures can be defined. As a consequence we obtain a new and noninductive proof of the existence of hierarchies of Poisson–Nijenhuis structures.

In this paper we study the super-Poisson structures on the Cartan algebra. The main result is that every super-Poisson structure on A(M) has associated a unique derivation from A(M) into the graded module of TM-valued differential forms and that any such derivation that satisfies certain conditions give raise forms super-Poisson structure.

Given a symplectic structure on a differentiable manifold, M, we show, by applying a characterization of derivations, that there exists an associated even graded symplectic structure on any graded manifold whose underlying manifold is M. We also show a similar result for odd graded symplectic structures.