Cartan Ribbonization of Surfaces and a Topological Inspection

We develop the concept of Cartan ribbons and a method by which they can be used to ribbonize any given surface in space by intrinsically flat ribbons. The geodesic curvature along the center curve on the surface agrees with the geodesic curvature of the corresponding Cartan development curve, and this makes a rolling strategy successful. Essentially, it follows from the orientational alignment of the two co-moving Darboux frames during the rolling. Using closed center curves we obtain closed approximating Cartan ribbons that contribute zero to the total curvature integral of the ribbonization. This paves the way for a particular simple topological inspection -- it is reduced to the question of how the ribbons organize their edges relative to each other. The Gauss-Bonnet theorem leads to this topological inspection of the vertices. Finally, we display two examples of ribbonizations of surfaces, namely of a torus using two ribbons, and of an ellipsoid using its closed curvature lines as center curves for the ribbons. The topological inspection of the torus ribbonization is particularly simple as it has no vertex points, giving directly the Euler characteristic $0$. The ellipsoid has $4$ vertices -- corresponding to the $4$ umbilical points -- each of degree one and each therefore contributing one-half to the Euler characteristic

An Improved FPT Algorithm for the Flip Distance Problem

Given a set cal P of points in the Euclidean plane and two triangulations of cal P, the flip distance between these two triangulations is the minimum number of flips required to transform one triangulation into the other. The Parameterized Flip Distance problem is to decide if the flip distance between two given triangulations is equal to a given integer k. The previous best FPT algorithm runs in time O^*(kcdot c^k) (cleq 2times 14^11), where each step has fourteen possible choices, and the length of the action sequence is bounded by 11k. By applying the backtracking strategy and analyzing the underlying property of the flip sequence, each step of our algorithm has only five possible choices. Based on an auxiliary graph G, we prove that the length of the action sequence for our algorithm is bounded by 2|G|. As a result, we present an FPT algorithm running in time O^*(kcdot 32^k)

An Integrated Procedure to Assess the Stability of Coastal Rocky Cliffs: From UAV Close-Range Photogrammetry to Geomechanical Finite Element Modeling

The present paper explores the combination of unmanned aerial vehicle (UAV) photogrammetry and three-dimensional geomechanical modeling in the investigation of instability processes of long sectors of coastal rocky cliffs. The need of a reliable and detailed reconstruction of the geometry of the cliff surfaces, beside the geomechanical characterization of the rock materials, could represent a very challenging requirement for sub-vertical coastal cliffs overlooking the sea. Very often, no information could be acquired by alternative surveying methodologies, due to the absence of vantage points, and the fieldwork could pose a risk for personnel. The case study is represented by a 600 m long sea cliff located at Sant\u2019Andrea (Melendugno, Apulia, Italy). The cliff is characterized by a very complex geometrical setting, with a suggestive alternation of 10 to 20 m high vertical walls, with frequent caves, arches and rock-stacks. Initially, the rocky cliff surface was reconstructed at very fine spatial resolution from the combination of nadir and oblique images acquired by unmanned aerial vehicles. Successively, a limited area has been selected for further investigation. In particular, data refinement/decimation procedure has been assessed to find a convenient three-dimensional model to be used in the finite element geomechanical modeling without loss of information on the surface complexity. Finally, to test integrated procedure, the potential modes of failure of such sector of the investigated cliff were achieved. Results indicate that the most likely failure mechanism along the sea cliff examined is represented by the possible propagation of shear fractures or tensile failures along concave cliff portions or over-hanging due to previous collapses or erosion of the underlying rock volumes. The proposed approach to the investigation of coastal cliff stability has proven to be a possible and flexible tool in the rapid and highly-automated investigation of hazards to slope failure in coastal areas

Smooth surface reconstruction and regularisation

In this paper, we address the problem of smooth surface reconstruction from image sequences obtained with known camera motion . An approach based on a local reconstruction method is presented . Previous works have shown that local shape estimation can b e achieved using the occluding contours . Several algorithms allow such a reconstruction and we present our approach . Recovered rim points are then used to compute a triangular mesh . Resulting meshes are regularised in order to correct noise perturbation s which affect the reconstruction . Experimentations on real data are shown to illustrate the whole process .Dans cet article, nous abordons le problème de la reconstruction de surface d'objets courbes à partir de séquences d'images. Nous présentons une approche basée sur une méthode de reconstruction locale. De précédents travaux ont montré qu'une reconstruction locale de la surface était possible à partir de ses contours occultants. Plusieurs algorithmes permettent une telle reconstruction et nous présentons notre approche. Les points ainsi reconstruits servent ensuite à construire un maillage triangulaire. Les maillages obtenus sont régularisés pour corriger les effets du bruit présent dansle processus de reconstruction. Des expérimentations réalisées sur des données réelles viennent illustrer l'ensemble de la méthode

Aproximació facetada de superfícies paramètriques retallades

Postprint (published version

Virtual sculpting : an investigation of directly manipulated free-form deformation in a virtual environment

This thesis presents a Virtual Sculpting system, which addresses the problem of Free-Form Solid Modelling. The disparate elements of a Polygon-Mesh representation, a Directly Manipulated Free-Form Deformation sculpting tool, and a Virtual Environment are drawn into a cohesive whole under the mantle of a clay-sculpting metaphor. This enables a user to mould and manipulate a synthetic solid interactively as if it were composed of malleable clay. The focus of this study is on the interactivity, intuitivity and versatility of such a system. To this end, a range of improvements is investigated which significantly enhances the efficiency and correctness of Directly Manipulated Free-Form Deformation, both separately and as a seamless component of the Virtual Sculpting system