Connecting the 3D DGS Calques3D with the CAS Maple

Many (2D) Dynamic Geometry Systems (DGSs) are able to export numeric coordinates and equations with numeric coefficients to Computer Algebra Systems (CASs). Moreover, different approaches and systems that link (2D) DGSs with CASs, so that symbolic coordinates and equations with symbolic coefficients can be exported from the DGS to the CAS, already exist. Although the 3D DGS Calques3D can export numeric coordinates and equations with numeric coefficients to Maple and Mathematica, it cannot export symbolic coordinates and equations with symbolic coefficients. A connection between the 3D DGS Calques3D and the CAS Maple, that can handle symbolic coordinates and equations with symbolic coefficients, is presented here. Its main interest is to provide a convenient time-saving way to explore problems and directly obtain both algebraic and numeric data when dealing with a 3D extension of “ruler and compass geometry”. This link has not only educational purposes but mathematical ones, like mechanical theorem proving in geometry, geometric discovery (hypotheses completion), geometric loci finding… As far as we know, there is no comparable “symbolic” link in the 3D case, except the prototype 3D-LD (restricted to determining algebraic surfaces as geometric loci).     

Calculation of phase equilibria for an alloy nanoparticle in contact with a solid nanowire

Phase equilibria in a system constituted of an alloy nanoparticle in contact with a solid nanowire have been modelled based on the minimization of a Gibbs free energy function. The Gibbs free energy consists of a bulk, surface and interface contribution. The bulk contribution is taken from CALPHAD thermodynamic databases and the surface properties from the literature. The effect of particle size and surface and interfacial properties on the liquidus line of the Au-Ge and In-Si systems has been studied. The results are compared to the bulk phase diagram and phase equilibria calculated for nano-systems with different geometries.     

Sampling methods for shortest vectors,closest vectors and successive minima

We study four problems from the geometry of numbers, the shortest vector problem  (Svp)$\left(\text{<small-caps>Svp</small-caps>}\right)$, the closest vector problem  (Cvp)$\left(\text{<small-caps>Cvp</small-caps>}\right)$, the successive minima problem  (Smp)$\left(\text{<small-caps>Smp</small-caps>}\right)$, and the shortest independent vectors problem   (Sivp$\text{<small-caps>Sivp</small-caps>}$). Extending and generalizing results of Ajtai, Kumar, and Sivakumar we present probabilistic single exponential time algorithms for all four problems for all _?p${?}_p$ norms. The results on Smp$\text{<small-caps>Smp</small-caps>}$ and Sivp$\text{<small-caps>Sivp</small-caps>}$ are new for all norms. The results on Svp$\text{<small-caps>Svp</small-caps>}$ and Cvp$\text{<small-caps>Cvp</small-caps>}$ generalize previous results of Ajtai et al. for the Euclidean _?2${?}_2$ norm to arbitrary _?p${?}_p$ norms. We achieve our results by introducing a new lattice problem, the generalized shortest vector problem   (GSvp$\text{<small-caps>GSvp</small-caps>}$). ¹ We describe a single exponential time algorithm for GSvp$\text{<small-caps>GSvp</small-caps>}$. We also describe polynomial time reductions from Svp,Cvp,Smp$\text{<small-caps>Svp</small-caps>},\text{<small-caps>Cvp</small-caps>},\text{<small-caps>Smp</small-caps>}$, and Sivp$\text{<small-caps>Sivp</small-caps>}$ to GSvp$\text{<small-caps>GSvp</small-caps>}$, establishing single exponential time algorithms for the four classical lattice problems. This approach leads to a unified algorithmic treatment of the lattice problems Svp,Cvp,Smp$\text{<small-caps>Svp</small-caps>},\text{<small-caps>Cvp</small-caps>},\text{<small-caps>Smp</small-caps>}$, and Sivp$\text{<small-caps>Sivp</small-caps>}$.     

Discrete Critical Values: a General Framework for Silhouettes Computation

Many shapes resulting from important geometric operations in industrial applications such as Minkowski sums or volume swept by a moving object can be seen as the projection of higher dimensional objects. When such a higher dimensional object is a smooth manifold, the boundary of the projected shape can be computed from the critical points of the projection. In this paper, using the notion of polyhedral chains introduced by Whitney, we introduce a new general framework to define an analogous of the set of critical points of piecewise linear maps defined over discrete objects that can be easily computed. We illustrate our results by showing how they can be used to compute Minkowski sums of polyhedra and volumes swept by moving polyhedra.     

Interactive Cover Design Considering Physical Constraints

We developed an interactive system to design a customized cover for a given three‐dimensional (3D) object such as a camera, teapot, or car. The system first computes the convex hull of the input geometry. The user segments it into several cloth patches by drawing on the 3D surface. This paper provides two technical contributions. First, it introduces a specialized flattening algorithm for cover patches. It makes each two‐dimensional edge in the flattened pattern equal to or longer than the original 3D edge; a smaller patch would fail to cover the object, and a larger patch would result in extra wrinkles. Second, it introduces a mechanism to verify that the user‐specified opening would be large enough for the object to be removed. Starting with the initial configuration, the system virtually “pulls” the object out of the cover while avoiding excessive stretching of cloth patches. We used the system to design real covers and confirmed that it functions as intended.     

Boundary Detection in Particle‐based Fluids

This paper presents a novel method to detect free‐surfaces on particle‐based volume representation. In contrast to most particle‐based free‐surface detection methods, which perform the surface identification based on physical and geometrical properties derived from the underlying fluid flow simulation, the proposed approach only demands the spatial location of the particles to properly recognize surface particles, avoiding even the use of kernels. Boundary particles are identified through a Hidden Point Removal (HPR) operator used for visibility test. Our method is very simple, fast, easy to implement and robust to changes in the distribution of particles, even when facing large deformation of the free‐surface. A set of comparisons against state‐of‐the‐art boundary detection methods show the effectiveness of our approach. The good performance of our method is also attested in the context of fluid flow simulation involving free‐surface, mainly when using level‐sets for rendering purposes.     

Buoyancy Optimization for Computational Fabrication

This paper introduces a design and fabrication pipeline for creating floating forms. Our method optimizes for buoyant equilibrium and stability of complex 3D shapes, applying a voxel‐carving technique to control the mass distribution. The resulting objects achieve a desired floating pose defined by a user‐specified waterline height and orientation. In order to enlarge the feasible design space, we explore novel ways to load the interior of a design using prefabricated components and casting techniques. 3D printing is employed for high‐precision fabrication. For larger scale designs we introduce a method for stacking lasercut planar pieces to create 3D objects in a quick and economic manner. We demonstrate fabricated designs of complex shape in a variety of floating poses.     

Space‐Time Co‐Segmentation of Articulated Point Cloud Sequences

Consistent segmentation is to the center of many applications based on dynamic geometric data. Directly segmenting a raw 3D point cloud sequence is a challenging task due to the low data quality and large inter‐frame variation across the whole sequence. We propose a local‐to‐global approach to co‐segment point cloud sequences of articulated objects into near‐rigid moving parts. Our method starts from a per‐frame point clustering, derived from a robust voting‐based trajectory analysis. The local segments are then progressively propagated to the neighboring frames with a cut propagation operation, and further merged through all frames using a novel space‐time segment grouping technqiue, leading to a globally consistent and compact segmentation of the entire articulated point cloud sequence. Such progressive propagating and merging, in both space and time dimensions, makes our co‐segmentation algorithm especially robust in handling noise, occlusions and pose/view variations that are usually associated with raw scan data.     

A Hierarchical Approach for Regular Centroidal Voronoi Tessellations

In this paper, we consider Centroidal Voronoi Tessellations (CVTs) and study their regularity. CVTs are geometric structures that enable regular tessellations of geometric objects and are widely used in shape modelling and analysis. While several efficient iterative schemes, with defined local convergence properties, have been proposed to compute CVTs, little attention has been paid to the evaluation of the resulting cell decompositions. In this paper, we propose a regularity criterion that allows us to evaluate and compare CVTs independently of their sizes and of their cell numbers. This criterion allows us to compare CVTs on a common basis. It builds on earlier theoretical work showing that second moments of cells converge to a lower bound when optimizing CVTs. In addition to proposing a regularity criterion, this paper also considers computational strategies to determine regular CVTs. We introduce a hierarchical framework that propagates regularity over decomposition levels and hence provides CVTs with provably better regularities than existing methods. We illustrate these principles with a wide range of experiments on synthetic and real models.     

Pairwise Registration by Local Orientation Cues

Inspired by recent work on robust and fast computation of 3D Local Reference Frames (LRFs), we propose a novel pipeline for coarse registration of 3D point clouds. Key to the method are: (i) the observation that any two corresponding points endowed with an LRF provide a hypothesis on the rigid motion between two views, (ii) the intuition that feature points can be matched based solely on cues directly derived from the computation of the LRF, (iii) a feature detection approach relying on a saliency criterion which captures the ability to establish an LRF repeatably. Unlike related work in literature, we also propose a comprehensive experimental evaluation based on diverse kinds of data (such as those acquired by laser scanners, Kinect and stereo cameras) as well as on quantitative comparison with respect to other methods. We also address the issue of setting the many parameters that characterize coarse registration pipelines fairly and realistically. The experimental evaluation vouches that our method can handle effectively data acquired by different sensors and is remarkably fast.