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International audience; Given three regular space curves r1(t), r2(t), r3(t) for t in [0,1] that define a curvilinear triangle, we consider the problem of constructing a triangular surface patch R(u1,u2,u3) bounded by these three curves, such that they are geodesics of the constructed surface. Results from a prior study (Farouki et al., 2009a) concerned with tensor-product patches are adapted to identify constraints on the given curves for the existence of such geodesic-bounded triangular surface patches. For curves satisfying these conditions, the patch is constructed by means of a cubically-blended triangular Coons interpolation scheme. A formulation of thin-plate spline energy in terms of barycentric coordinates with respect to a general domain triangle is also derived, and used to optimize the smoothness of the geodesic-bounded triangular surface patches.

International audience; Given four polynomial or rational Bézier curves defining a curvilinear rectangle, we consider the problem of constructing polynomial or rational tensor-product Bézier 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 Bézier 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.

We present two numerical methods to approximate the shortest path or a geodesic between two points on a three-dimensional parametric surface. The first one consists of minimizing the path length, working in the parameter domain, where the approximation class is composed of Bezier curves. In the second approach, we consider Bezier surfaces and their control net. The numerical implementation is based on finding the shortest path on the successive control net subdivisions. The convergence property of the Bezier net to the surface gives an approximation of the required shortest path. These approximations, also called pseudo-geodesics, are then applied to the creation of pseudo-geodesic meshes. Experimental results are also provided.

International audience; 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.

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.

International audience; 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 RRFM 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 Bézier/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.

International audience; This paper is concerned with reconstruction of numerical or real surfaces based on the knowledge of some geodesic curves on the surface. So, considering two regular 3D-curves f0(t) and f1(t), our purpose is to construct a surface which interpolates these two curves in such a way that these two curves are geodesics on this surface. This will be accomplished using Hermite interpolation. For a real surface, it will be shown that geodesics can be acquired using a ribbon of micro-sensors.

International audience; 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.

We presented different problems in separation theory and showed the simplicity of some geometrical solutions in the particular case we have named “regular selectivity”. Using the same geometrical framework, extensions of these solutions to some more complicated cases, will be the aim of future works.