Extraction of cylinders and cones from minimal point sets

We propose new algebraic methods for extracting cylinders and cones from minimal point sets, including oriented points. More precisely, we are interested in computing efficiently cylinders through a set of three points, one of them being oriented, or through a set of five simple points. We are also interested in computing efficiently cones through a set of two oriented points, through a set of four points, one of them being oriented, or through a set of six points. For these different interpolation problems, we give optimal bounds on the number of solutions. Moreover, we describe algebraic methods targeted to solve these problems efficiently

Extraction of tori from minimal point sets

International audienceA new algebraic method for extracting tori from a minimal point set, made of two oriented points and a simple point, is proposed. We prove a degree bound on the number of such tori; this bound is reached on examples, even when we restrict to smooth tori. Our method is based on pre-computed closed formulae well suited for numerical computations with approximate input data

Constructive Lattice Geometry

Lattice structures are widespread in product and architectural design. Recent work has demonstrated the printing of nano-scale lattices. However, an anticipated increase in product complexity will require the storage, processing, and design of lattices with orders of magnitude more elements than current Computer-Aided Design (CAD) software can manage. To address this, we propose a class of highly regular lattices called Steady Lattices, which due to their regularity, provide opportunities for highly compressed storage, accelerated processing, and intuitive design. Special cases of steady lattices are also presented, which provide varying degrees of compromise between design freedom and geometric regularity. For example, the commonly used regular lattices, which provide little design freedom but offer maximum regularity, are the least general form of steady lattice. We propose the 2-directional, Bent Corner-Operated Trans-Similar (BeCOTS) lattices as a useful compromise between regular lattices and fully general steady lattices. A BeCOTS lattice may be controlled by four non-coplanar points, which represent four corners of the lattice. The Trans-Similar property ensures that a BeCOTS lattice is composed of groups of beams such that each consecutive pair of groups of beams along a particular direction is related by the same similarity. Trans-Similarity also enables constant-time queries such as surface area calculation, volume calculation, and point-membership classification. We take advantage of the regularity in steady lattices to efficiently produce and query highly complex lattice structures that we call Constructive Lattice Geometry (CLG), where CLG is an extension of traditional Constructive Solid Geometry (CSG). CLG models are periodic CSG models for which regular patterns of primitives are combined into many repeating CSG microstructures that are ultimately combined into one CSG macrostructure. We provide strategies for designing and processing recursively defined CLG models to enable the creation of CLG models composed of smaller CLG models. Parameterized steady lattices and CLG models may be defined by a few lines of code, which facilitates lazy (on-demand) evaluation, massively parallel processing, interactive editing, and algorithmic optimization.Ph.D