Search results for "implicit function"

Rational supershapes for surface reconstruction

2007

Simple representation of complex 3D data sets is a fundamental problem in computer vision. From a quality control perspective, it is crucial to use efficient and simple techniques do define a reference model for further recognition or comparison tasks. In this paper, we focus on reverse engineering 3D data sets by recovering rational supershapes to build an implicit function to represent mechanical parts. We derive existing techniques for superquadrics recovery to the supershapes and we adapt the concepts introduced for the ratioquadrics to introduce the rational supershapes. The main advantage of rational supershapes over standard supershapes is that the radius is now expressed as a ration…

Regular k-Surfaces

2012

Roughly speaking, a regular surface in \(\mathbb{R}^3\) is a two-dimensional set of points, in the sense that it can be locally described by two parameters (the local coordinates) and with the property that it is smooth enough (that is, there are no vertices, edges, or self-intersections) to guarantee the existence of a tangent plane to the surface at each point.

INITIAL PARAMETRIC REPRESENTATION OF BLOBS

2009

Blobs, developed by J.F. Blinn in 1982, are the implicit surfaces obtained by composition of a real numerical function and a distance function. Since, many authors (C. Murakami, H. Nishimura, G. Wyvill…) defined their own function of density, from these implicit surfaces are interesting from several points of view. In particular, their fusion makes it possible to easily obtain an implicit equation of resulting surface. However, these surfaces do not admit a parametric equation yet. In this article, we will establish the parametric equation of two blobs in fusion, defined by the function of density of C. Murakami, by using an algebraic method. Then, we will develop another method, based on …

Boolean operations with implicit and parametric representation of primitives using R-functions

2005

We present a new and efficient algorithm to accurately polygonize an implicit surface generated by multiple Boolean operations with globally deformed primitives. Our algorithm is special in the sense that it can be applied to objects with both an implicit and a parametric representation, such as superquadrics, supershapes, and Dupin cyclides. The input is a constructive solid geometry tree (CSG tree) that contains the Boolean operations, the parameters of the primitives, and the global deformations. At each node of the CSG tree, the implicit formulations of the subtrees are used to quickly determine the parts to be transmitted to the parent node, while the primitives' parametric definition …

A NEW POTENTIAL FUNCTION FOR SELF INTERSECTING GIELIS CURVES WITH RATIONAL SYMMETRIES

2009

International audience; We present a new potential field equation for self-intersecting Gielis curves with rational rotational symmetries. In the literature, potential field equations for these curves, and their extensions to surfaces, impose the rotational symmetries to be integers in order to guarantee the unicity of the intersection between the curve/surface and any ray starting from its center. Although the representation with natural symmetries has been applied to mechanical parts modeling and reconstruction, the lack of a potential function for Rational symmetry Gielis Curves (RGC) remains a major problem for natural object representation, such as flowers and phyllotaxis. We overcome thi…

On some bifurcation analysis techniques for continuous systems

2016

This paper is devoted to techniques in bifurcation analysis for continuous mechanical systems, concentrating on polynomial equations and implicitly given functions. These are often encountered in problems of mechanics and especially in stability analysis. Taking a classical approach, we summarize the relevant features of the cubic polynomial equation, and present some new aspects for asymptotics and parametric representation of the solutions. This is followed by a brief look into the implicit function theorem as a tool for analyzing bifurcations. As an example from mechanics, we consider bifurcations in the transverse free vibration problem of an axially compressed beam. peerReviewed

The historical role played by the work of Ulisse Dini on implicit function theory in the modern differential geometry foundations: the case of the st…

2012

On the History of Differentiable Manifolds

2012

International audience; We discuss central aspects of history of the concept of an affine dif-ferentiable manifold, as a proposal confirming the need for using some quantitative methods (drawn from elementary Model Theory) in Mathematical Historiography. In particular, we prove that this geometric structure is a syntactic rigid designator in the sense of Kripke-Putnam.

A Complete, Exact and Efficient Implementation for Computing the Edge-Adjacency Graph of an Arrangement of Quadrics

2011

International audience; We present a complete, exact and efficient implementation to compute the edge-adjacency graph of an arrangement of quadrics, i.e. surfaces of algebraic degree 2. This is a major step towards the computation of the full 3D arrangement. We enhanced an implementation for an exact parameterization of the intersection curves of two quadrics, such that we can compute the exact parameter value for intersection points and from that the edge-adjacency graph of the arrangement. Our implementation is complete in the sense that it can handle all kinds of inputs including all degenerate ones, i.e. singularities or tangential intersection points. It is exact in that it always comp…