Marie-Laurence Mazure

Mathematical methods for curves and surfaces

We will deal with the translation surfaces which are the shapes generated by translating one curve along another one. We focus on the geometry of translation surfaces generated by two algebraic curves in space and study their properties, especially those useful for geometric modelling purposes. It is a classical result that each minimal surface may be obtained as a translation surface generated by an isotropic curve and its complex conjugate. Thus, we can study the minimal surfaces as special instances of translation surfaces. All the results about translation surfaces will be directly applied also to minimal surfaces. Finally, we present a construction of rational isotropic curves with a prescribed tangent field which leads to the description of all rational minimal surfaces. A close relation to surfaces with Pythagorean normals will be also discussed.

Blossoms and Optimal Bases

It is now classical to define blossoms by means of intersections of osculating flats. We consider here the most general context of spline spaces with sections in arbitrary extended Chebyshev spaces and with connections defined by arbitrary lower triangular matrices with positive diagonal elements. We show how the existence of blossoms in such spaces automatically leads to optimal bases in the sense of Carnicer and Peña.

Chebyshev-Bernstein bases

Abstract From a design point of view, extended Chebyshev spaces are interesting in so far as they provide useful shape parameters. In such spaces, the blossoming principle generates de Casteljau type algorithms, which makes it natural to define Chebyshev–Bernstein bases as the dual bases of the linear functional giving the control points. Such bases share the same properties as the Bernstein bases in polynomial spaces. In particular they are the optimal shape preserving bases. This also holds in piecewise smooth extended Chebyshev spaces.

Ready-to-Blossom Bases in Chebyshev Spaces

Abstract This paper gives a survey on blossoms and Chebyshev spaces, with a number of new results and proofs. In particular, Extended Chebyshev spaces are characterised by the existence of a certain type of bases which are especially suited to enable us to prove both existence and properties of blossoms under the weakest possible differentiability assumptions. We also examine the case of piecewise spaces built from different Extended Chebyshev spaces and connection matrices.

Chebyshev splines beyond total positivity

For polynomial splines as well as for Chebyshev splines, it is known that total positivity of the connection matrices is sufficient to obtain B-spline bases. In this paper we give a necessary and sufficient condition for the existence of B-bases (or, equivalently, of blossoms) for splines with connection matrices and with sections in different four-dimensional extended Chebyshev spaces.

How to build all Chebyshevian spline spaces good for geometric design

In the present work we determine all Chebyshevian spline spaces good for geometric design. By Chebyshevian spline space we mean a space of splines with sections in different Extended Chebyshev spaces and with connection matrices at the knots. We say that such a spline space is good for design when it possesses blossoms. To justify the terminology, let us recall that, in this general framework, existence of blossoms (defined on a restricted set of tuples) makes it possible to develop all the classical geometric design algorithms for splines. Furthermore, existence of blossoms is equivalent to existence of a B-spline bases both in the spline space itself and in all other spline spaces derived from it by insertion of knots. We show that Chebyshevian spline spaces good for design can be described by linear piecewise differential operators associated with systems of piecewise weight functions, with respect to which the connection matrices are identity matrices. Many interesting consequences can be drawn from the latter characterisation: as an example, all Chebsyhevian spline spaces good for design can be built by means of integral recurrence relations.

Finding all systems of weight functions associated with a given extended Chebyshev space

Systems of weight functions and corresponding generalised derivatives are classically used to build extended Chebyshev spaces on a given interval. This is a well-known procedure. Conversely, if the interval is closed and bounded, it is known that a given extended Chebyshev space can always be associated with a system of weight functions via the latter procedure. In the present article we determine all such possibilities, that is, all systems of weight functions which can be used to define a given extended Chebyshev space on a closed bounded interval.

Quasi-Chebyshev splines with connection matrices: application to variable degree polynomial splines

Extending a result recently obtained for Chebyshevian splines, we give a necessary and sufficient condition for the existence of blossoms (or, equivalently, of B-spline bases) for splines with connection matrices and with sections in different four-dimensional quasi-Chebyshev spaces. We apply this result to the study of variable degree polynomial splines.

Bernstein-type operators in Chebyshev spaces

We prove that it is possible to construct Bernstein-type operators in any given Extended Chebyshev space and we show how they are connected with blossoms. This generalises and explains a recent result by Aldas/Kounchev/Render on exponential spaces. We also indicate why such operators automatically possess interesting shape preserving properties and why similar operators exist in still more general frameworks, e.g., in Extended Chebyshev Piecewise spaces. We address the problem of convergence of infinite sequences of such operators, and we do prove convergence for special instances of Müntz spaces.

Chebyshev spaces with polynomial blossoms

The use of extended Chebyshev spaces in geometric design is motivated by the interesting shape parameters they provide. Unfortunately the algorithms such spaces lead to are generally complicated because the blossoms themselves are complicated. In order to make up for this inconvenience, we here investigate particular extended Chebyshev spaces, containing the constants and power functions whose exponents are consecutive positive integers. We show that these spaces lead to simple algorithms due to the fact that the blossoms are polynomial functions. Furthermore, we also describe an elegant dimension elevation algorithm which makes it possible to return to polynomial spaces and therefore to use all the classical algorithms for polynomials.