Section: New Results

Geometry

Geometric modeling with quadric patches

Participants : Bernard Mourrain, Changhe Tu [ Shandong Univ. ] , Jiaye Wang [ Shandong Univ. ] , Wenping Wang [ Hong Kong Univ. ] .

Representing shapes by non-linear model is a challenge in geometric modeling, which aims at developping geometric computation on compact and rich models. Our work on quadric patches is related to this objective. We are interested in developping basic algebraic-geometric operations which are required for exact and efficient manipulations of quadrics in this context. One of the key operation is the intersection of quadrics. We present an efficient method for classifying the morphology of the intersection curve of two quadrics (QSIC) in , 3D real projective space; including an analysis of singularity, reducibility, the number of connected components, and the degree of each irreducible component, etc. There are in total 35 different QSIC morphologies with non-degenerate quadric pencils, which we characterize by by algebraic condition expressed in terms of the Segre characteristics and the signature sequence of a quadric pencil. We show how to compute a signature sequence with rational arithmetic so as to determine the morphology of the intersection curve of any two given quadrics. Two immediate applications of our results are the robust topological classification of QSIC in computing B-rep surface representation in solid modeling and the derivation of algebraic conditions for collision detection of quadric primitives. This work is submitted for a journal publication.

A computational study of ruled surfaces

Participants : Laurent Busé, Marc Dohm, André Galligo.

Implicitization is a fundamental problem in Computer Aided Geometric Design and there are numerous applications related to it, e.g. the computation of the intersection of two ruled surfaces. The method of -bases (also known as "moving lines" or "moving surfaces") constitutes an efficient solution to the implicitization problem. Introduced in 1998 by Cox, Sederberg, and Chen for rational curves, it was generalized later to ruled surfaces. Whereas the curve case is very well understood and we know that the resultant of a -basis is the implicit equation to the power d, where d is the degree of the parametrization, this result is still to be shown in its full generality (i.e. for arbitrary d) for ruled surfaces. This work filled this gap by giving a proof, which relies on a geometric idea that reduces the ruled surface case to the curve case. This work has been presented at the EACA conference and is submitted for publication in a journal.

General classification of (1,2) parametric surfaces in

Participants : Thi Ha Lé, André Galligo.

Patches of parametric real surfaces of low degrees are commonly used in Computer Aided Geometric Design and Geometric Modeling. However the precise description of the geometry of the whole real surface is generally difficult to master, and few complete classifications exist. We study surfaces of bidegree (1,2). We present a classification and a geometric study of parametric surfaces of bidegree (1,2) over the complex field and over the real field by considering a dual scroll. We detect and describe (if it is not void) the trace of self-intersection and singular locus in the system of coordinates attached to the control polygon of a patch (1,2) in the box [0;1]×[0;1] . This work has been accepted for publication.

Using Bézier patches of bidegree (1,2) for corn leaf modeling

Participants : Franck Aries [ INRA of Avignon ] , André Galligo, Thi Ha Lé.

Surface modeling has a wide range of applications in botany including canopy models for teledetection, computation of radiative balance and optical properties. The detailed description of the architecture of vegetation canopies is critical for the modeling of many processes. The area, size, shape, orientation and position of the leaves drive the efficiency with which exchanges will occur with the atmosphere, including gas such as water vapor or carbon dioxide, liquids such as water, and radiation. Up to now, models describing canopy functioning are based on very simple approximations of the canopy architecture. A more detailed description is required when investigations focus on the role of particular organs in the canopy, the description of rain interception, the propagation of diseases from organs to others, or the radiative transfer to describe the reflected or emitted fluxes in a range of wavelengths. A corn leaf, as well as many leaves, is formed by a nervure separating 2 ruled surface patches. We investigate such representations with patches of bidegree (1,2). Our approach consists in representing the nervure as a conic or a cubic spline curve, relying on the fact that a line in the parameter space is mapped on a twisted cubic by a parameterization of bidegree (1,2). Then we choose (1,2) Bézier patches delimited by this segment of curve to model the leaf. We present a tool box of algorithms adapted to our purpose, for the patches of bidegree (1,2). This includes fast detection algorithm for selfintersection and intersection, area estimation and computation of normals. In a following paper with a more botany flavor, we will present a realistic model for modeling a corn parcel.

Biquadratic sampling method for intersection of surfaces

Participants : Stéphane Chau, André Galligo, Jean-Pascal Pavone.

In Computer Aided Geometric Design (CAGD), the parameterized surfaces are used delimiting volumes. The computation of the intersection curve between such two surfaces is thus crucial for the description of the CAGD objects. An often used method to address this problem consist in using a mesh for each surface, and then to proceed to their intersection (intersection of triangles). Other methods for the intersection problem deal with global representations of the surfaces such as B-splines, however the usual CAGD procedures (offsetting, drafting, ...) do not conserve this model and procedural surfaces that are based on approximations are required. So, even if the intersection methods are exact, they only provide an approximation of the "real" intersection curve. This can be of bad quality, because the approximations of the surfaces are separated from the intersection process and are somehow "global". In the general case, the only informations that we can access for a parameterized surface are "local" (its evaluation). Thus, a general algorithm is a sampling algorithm. Moreover, in order to design a more robust algorithm which does not avoid possible intersection between two parameterized surfaces S₁ and S₂ , we consider the local extreme values of the distance function between S₁ and S₂ . We are interested in an intersection method based on approximation by biquadratic patches. It can be seen as a good compromise between the approximation by B-splines and meshes because it is a local approximation (like the meshes case) and the distance function can have extreme values (like the B-spline case). It can be used for the detection of touching points, and bifurcation points which are undetectable by a mesh approximation. This method is almost as efficient as a classical sampling method, but it is more robust. Moreover, this approach requires less evaluations than a classical sampling method, so it could be more interesting for complicated procedural surfaces. Our approach is based on two steps. At first, we build quadtrees hierarchy covering the grids of biquadratic patches and then we have a family of intersection problem between two biquadratic patches. This last problem was treated in work that has been accepted for publication but we also introduce a subdivision method to find the topology of the intersection curves.

Subdivision methods for 2d and 3d implicit curves

Participants : Chen Liang [ Univ. of Hong Kong ] , Bernard Mourrain, Jean-Pascal Pavone.

We describe a subdivision method for handling algebraic implicit curves in 2d and 3d. We use the representation of polynomials in the Bernstein basis associated with a given box, to check if the topology of the curve is determined inside this box, from its points on the border of the box. Subdivision solvers are involved for computing these points on the faces of the box, and segments joining these points are deduced to get a graph isotopic to the curve. Using envelop of polynomials, we show how this method allow to handle efficiently and accurately implicit curves with large coefficients. We report on implementation aspects and experimentations on 2d curves such as ridge curves or self intersection curves of parameterized surfaces, and on silhouette curves of implicit surfaces, showing the interesting practical behavior of this approach.

Intersection and self-intersection of surfaces by means of Bezoutian matrices

Participants : Laurent Busé, Mohamed Elkadi, André Galligo.

The computation of intersection and self-intersection loci of parameterized surfaces is an important task in Computer Aided Geometric Design and computer algebra tools need to be developed further for computing their implicit equations. In this work, we addressed these problems via four resultants with separated variables: two of them are specializations of more general ones and the others are determinantal. We gave a rigorous study in these cases and provide new and useful formulas via adapted computations of Bezoutians. A paper has been submitted for publication.

Topology of curves and surfaces

Participants : Lionel Alberti, George Comte [ UNSA ] , Bernard Mourrain, Jean-Pierre Técourt.

In order to produce a triangulation of an implicit surface, we look at subdivision methods. Our approach to the problem is through singularity theory. By Thom-Mather transversality theorem (which we can guarantee in a computable way) we can prove that in given domains, the surface is topologically trivial (that is, a product of its link with Rⁿ ). We can then recursively apply the method to the link of the surface. The classical method uses a CAD decomposition that even in its lighter forms has a complexity depending on the complexity of the projection making the complexity somewhat unrelated to the actual geometry of the surface. With this approach we do not have to rely on a projection and it is hoped that it will actually lead to a practically more efficient algorithm. This work has been submitted for publication. The approach behind this work is described in [12] . Some algebraic issues related to these geometric problems are discussed in [16] .

Intersection of algebraic surfaces

Participants : Daouda N'Diatta, Bernard Mourrain, Olivier Ruatta [ Univ. Limoges ] .

Numerical modeling plays an increasingly role in fields at the border between data processing and mathematics. This is the case for example in CAD (computer-aided design), where the objects of a scene or a piece to be built are represented by parameterized curves or surfaces such as NURBS, robotics (problem of the parallel robot, or vision), or molecular biology (rebuilding of a molecule starting from the matrix of the distances between its atoms obtained by NMR). A fundamental operator in this context is the intersection of geometric models, which leads to algebraic questions.

We focus on the problem of computing the topology of a plane algebraic curves and three-dimensional algebraic curves defined as the intersection of two algebraic surfaces. In the case of two parameterized surfaces, in order to reduce to such a situation, we may compute the implicit equation of one of the rational surfaces. This reduce the problem of intersection to the case of a curve defined by an implicit equation in the plane of parameters. Our main concern will be the case of implicit curves, either in the plane or in a three-dimensional space.

The problem of computing the topology of an algebraic curve received a lot of attention in the past literature. Different methods have been experimented, but they suffer from the problem of certifying the topology of the result. At first, we propose an algorithm based on sub-resultant to compute with certainty a simplicial complex isotopic to a plane algebraic curve defined by his implicit equation. Secondly, by using the 2D approach we present a new method to compute with certainty the topology of an algebraic curve in 3D defined by two polynomial equations even when the ideal generate by the two polynomial is not reduced.

These algorithms are beeing implemented in the synaps library.

Resolution of singularities

Participants : Lionel Alberti, Edward Bierstone [ Univ. Toronto ] .

This was done in collaboration with Edward Bierstone, Professor at the Fields Institute, Toronto, Canada.

The local properties of singularities are best understood looking at their resolution and blow-up maps. The structure of resolution maps is not well understood in arbitrary dimension, but we can get a rather good handle on what's happening in dimension 3 for isolated singularities. Resolution is to be understood as a mathematical tool enabling a better handling and understanding of singularities rather than as a goal in itself that will yield efficient algorithms.

Approximate implicitization

Participants : Stéphane Chau, André Galligo.

In order to approximate a general parameterized surface (given by evaluations) by biquadratic patches, we were interested in the approximate implicitization of these polynomial surfaces. The implicitization problem consists in finding the exact algebraic representation of a given rational parametric curve or surface. The classical methods to deal with this problem use the resultants and Groebner bases. But these algebraic approaches are very time-consuming (because of too high degrees) and have the drawback of introducing too many components in excess (so-called "phantom components"). So a very simple patch of surface can have a complicated exact implicit equation because the parameterization is local whereas the algebraic representation is global. Thus, instead of finding the exact representation of a given biquadratic patch, we can wonder how well a given algebraic surface approximates the corresponding parametric surface of bi-degree (2, 2) in the region of interest. This is the approximate implicitization problem and an algebraic approach based on factorization was developed by SINTEF. Our approach consists in the construction of some specific geometric constraints to give a linear system of equations and we use a singular value decomposition to solve it.

Dynamic and generic method for computing an arrangement of implicit curves

Participants : Julien Wintz, Bernard Mourrain.

Arrangements are very important issues in many application fields and have been studied for several years with different points of view. Arrangements are mainly computed using sweep line methods, such as the well known Bentley-Ottman algorithm for computing an arrangement of line segments. These methods are strongly related to the nature of objects and are not well suited for non-trivial objects such as closed curves or surfaces. We propose an arrangement algorithm which is generic in the sense that it is not related to the nature of the objects to be arranged. The algorithm is dynamic, which means, it can be maintained under insertion and deletions of the objects. Objects are always considered with a bounding domain so that we can also consider non convex objects. This domain is subdivided further if the cell is not deemed relevant, what is ensured by a regularity test which depends on the nature of input objects. A cell is relevant if regions can be deduced from the topology of the object within the cell. This general framework has been first tested for computing an arrangement of implicit curves. Type related elements of the algorithm are then specialized for implicit curves. These elements are regularity test and regions computation. A segment of implicit curve is said to be regular in a cell if it is either x-monotonic, y-monotonic or if it contains one and only one singular points and if the number of branches stemming out from this singularity is exactly the number of intersections of the curves with the cell. To compute the number of branches stemming out from a singularity, we use degree theory. To compute the intersection of the curve with the cell border (as well as the topological degree), we use the Bernstein conversion of the polynomials representing the curve. The arrangement is represented by an augmented influence graph on regions obtained by subdivision and merging steps. This algorithm is implemented within the algebraic modeler Axel and will quite easily be extended to the case of BSpline curves [19] .