Nicolas Szafran

Construction of Bézier surface patches with Bézier curves as geodesic boundaries

Given four polynomial or rational Bezier curves defining a curvilinear rectangle, we consider the problem of constructing polynomial or rational tensor-product Bezier patches bounded by these curves, such that they are geodesics of the constructed surface. The existence conditions and interpolation scheme, developed in a general context in earlier studies, are adapted herein to ensure that the geodesic-bounded surface patches are compatible with the usual polynomial/rational representation schemes of CAD systems. Precise conditions for four Bezier curves to constitute geodesic boundaries of a polynomial or rational surface patch are identified, and an interpolation scheme for the construction of such surfaces is presented when these conditions are satisfied. The method is illustrated with several computed examples.

Construction of rational surface patches bounded by lines of curvature

The fact that the Darboux frame is rotation-minimizing along lines of curvature of a smooth surface is invoked to construct rational surface patches whose boundary curves are lines of curvature. For given patch corner points and associated frames defining the surface normals and principal directions, the patch boundaries are constructed as quintic RRMF curves, i.e., spatial Pythagorean-hodograph (PH) curves that possess rational rotation-minimizing frames. The interior of the patch is then defined as a Coons interpolant, matching the boundary curves and their associated rotation-minimizing frames as surface Darboux frames. The surface patches are compatible with the standard rational Bezier/B-spline representations, and G^1 continuity between adjacent patches is easily achieved. Such patches are advantageous in surface design with more precise control over the surface curvature properties.

Reconstruction of quasi developable surfaces from ribbon curves

This paper deals with the acquisition and reconstruction of physical surfaces by mean of a ribbon device equipped with micro-sensors, providing geodesic curves running on the surface. The whole process involves the reconstruction of these 3D ribbon curves together with their global treatment so as to produce a consistent network for the geodesic surface interpolation by filling methods based on triangular Coons-like approaches. However, the ribbon curves follow their own way, subdividing thus the surface into arbitrary n-sided patches. We present here a method for the reconstruction of quasi developable surfaces from such n-sided curvilinear boundary curves acquired with the ribbon device.

Geometrical Modelling of the Fibre Organization in the Human Left Ventricle

The aim of the present study is to check, by means of elementary mathematical tools, a conjecture according to which myocardial fibres are geodesic curves running on some surfaces. This conjecture was first stated and experimentally checked by Streeter (1979) for the equatorial part of the left ventricle free wall. Quantitative polarized light microscopy provides measurements on fibre orientation that could lead to evidence that the conjecture remains true for the whole of the left ventricle. Study of the right ventricle is under progress.

GEOMETRIC MODELLING AND ALGORITHMS FOR BINARY MIXTURES

We present some computational methods in a particular case of mixing and separation theory, as an application of classical results in the field of computational geometry. Our aim is the production of some given mixtures by mixing parts of basic products. In the geometrical approach we use, products or mixtures are characterized by vectors and the mixing process by vector sums, in the vector space of physico-chemical species. The feasibility of a mixture is viewed as a point-inclusion in the convex set of mixtures which is a zonotope associated with basic mixtures. This paper is concerned with binary mixtures characterized by two species. The geometrical approach leads to plane geometry problems and gives complete solutions to the optimal fabrication of a mixture, as well as to the fabrication of a sequence of several mixtures. Using the framework of computational geometry, we present efficient algorithms for solving the main problems related to the management of binary mixtures.