A New Class of Curves of Rational B-Spline Type

A new class of rational parametrization has been developed and it was used to generate a new family of rational k functions B-splines which depends on an index α ∈ ]−∞ , 0[ ∪ ]1 , +∞[. This family of functions verifies, among other things, the properties of positivity, of partition of the unit and, for a given degree k, constitutes a true basis approximation of continuous functions. We loose, however, the regularity classical optimal linked to the multiplicity of nodes, which we recover in the asymptotic case, when α → ∞. The associated B-splines curves verify the traditional properties particularly that of a convex hull and we see a certain “conjugated symmetry” related to α. The case of open knot vectors without an inner node leads to a new family of rational Bezier curves that will be separately, object of in-depth analysis.

Enumeration of Planar Constellations with an Alternating Boundary

A planar hypermap with a boundary is defined as a planar map with a boundary, endowed with a proper bicoloring of the inner faces. The boundary is said alternating if the colors of the incident inner faces alternate along its contour. In this paper we consider the problem of counting planar hypermaps with an alternating boundary, according to the perimeter and to the degree distribution of inner faces of each color. The problem is translated into a functional equation with a catalytic variable determining the corresponding generating function. In the case of constellations—hypermaps whose all inner faces of a given color have degree $m\geq 2$, and whose all other inner faces have a degree multiple of $m$—we completely solve the functional equation, and show that the generating function is algebraic and admits an explicit rational parametrization. We finally specialize to the case of Eulerian triangulations—hypermaps whose all inner faces have degree $3$—and compute asymptotics which are needed in another work by the second author, to prove the convergence of rescaled planar Eulerian triangulations to the Brownian map.

Covering Rational Surfaces with Rational Parametrization Images

Let S be a rational projective surface given by means of a projective rational parametrization whose base locus satisfies a mild assumption. In this paper we present an algorithm that provides three rational maps f,g,h:A2⇢S⊂Pn such that the union of the three images covers S. As a consequence, we present a second algorithm that generates two rational maps f,g˜:A2⇢S, such that the union of its images covers the affine surface S∩An. In the affine case, the number of rational maps involved in the cover is in general optimal.

REAL ALGEBRAIC KNOTS OF LOW DEGREE

In this paper, we study rational real algebraic knots in ℝP3. We show that two real rational algebraic knots of degree ≤ 5 are rigidly isotopic if and only if their degrees and encomplexed writhes are equal. We also show that any smooth irreducible knot which admits a plane projection with less than or equal to four crossings has a rational parametrization of degree ≤6. Furthermore an explicit construction of rational knots of a given degree with arbitrary encomplexed writhe (subject to natural restrictions) is presented.