Juan Monterde

Bézier surfaces of minimal area: the Dirichlet approach

The Plateau-Bezier problem consists in finding the Bezier surface with minimal area from among all Bezier surfaces with prescribed border. An approximation to the solution of the Plateau-Bezier problem is obtained by replacing the area functional with the Dirichlet functional. Some comparisons between Dirichlet extremals and Bezier surfaces obtained by the use of masks related with minimal surfaces are studied.

On harmonic and biharmonic Bézier surfaces

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 Bezier surfaces. The main result we report here is that any biharmonic Bezier surface is fully determined by the boundary control points. We compare the new method, by way of practical examples, with some related methods such as surfaces generation using discretisation masks and functional minimisations.

A general 4th-order PDE method to generate Bézier surfaces from the boundary

In this paper we present a method for generating Bezier surfaces from the boundary information based on a general 4th-order PDE. This is a generalisation of our previous work on harmonic and biharmonic Bezier surfaces whereby we studied the Bezier solutions for Laplace and the standard biharmonic equation, respectively. Here we study the Bezier solutions of the Euler-Lagrange equation associated with the most general quadratic functional. We show that there is a large class of fourth-order operators for which Bezier solutions exist and hence we show that such operators can be utilised to generate Bezier surfaces from the boundary information. As part of this work we present a general method for generating these Bezier surfaces. Furthermore, we show that some of the existing techniques for boundary based surface design, such as Coons patches and Bloor-Wilson PDE method, are indeed particular cases of the generalised framework we present here.

Salkowski curves revisited: A family of curves with constant curvature and non-constant torsion

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 relation between these curves and rational curves with double Pythagorean hodograph is studied. A method to construct closed curves, including knotted curves, of constant curvature and continuous torsion using pieces of Salkowski curves is outlined.

Bézier Surfaces of Minimal Area

There are minimal surfaces admitting a Bezier 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 Ennepers surface.

Existence and uniqueness of solutions to superdifferential equations

Abstract 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: R 1¦1 x (M, AM) → (M, AM), satisfying a given initial condition. A necessary and sufficient condition for this integral flow to yield an R 1¦1-action is obtained: the homogeneous components, X0, and, X1, of the given field must define a Lie superalgebra of dimension (1, 1). The supergroup structure on R 1¦1, however, has to be specified: there are three non-isomorphic Lie supergroup structures on R 1¦1, all of which have addition as the group operation in the underlying Lie group R . On the other extreme, even if X0, and X1 do not close to form a Lie superalgebra, the integral flow of X is uniquely determined and is independent of the Lie supergroup structure imposed on R 1¦1. This fact makes it possible to establish an unambiguous relationship between the algebraic Lie derivative of supergeometric objects (e.g., superforms), and its geometrical definition in terms of integral flows. It is shown by means of examples that if a supergroup structure in R 1¦1 is fixed, some flows obtained from left-invariant supervector fields on Lie supergroups may fail to define an R 1¦1-action of the chosen structure. Finally, necessary and sufficient conditions for the integral flows of two supervector fields to commute are given.

The Plateau-Bézier Problem

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

Jacobi-Nijenhuis manifolds and compatible Jacobi structures

Abstract 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.

A characterization of helical polynomial curves of any degree

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.

A characterization of graded symplectic structures

Abstract 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. These graded symplectic forms give rise to canonical Poisson brackets on the graded manifold.