# Project : galaad

## Section: New Results

### Geometry

#### Geometric modeling and Steiner surfaces

Participants : Franck Aries [INRA Avignon], Jean-Pierre Técourt, Bernard Mourrain.

We study quadratic parameterization patches (generically Steiner patches) for modeling purpose. We give an overview of the properties and algorithmic of these surfaces, including classification, implicitization, and inversion. Our new approach for the classification is based on rational and certified computation, using effective algebraic geometry tools. We use resultant-based constructions to deduce the implicit equations of such patches. Exploiting the properties of these resultant matrices, we propose a new approach to solve the inversion problem. We show how it reduces to a linear algebra problem, for which we propose a stable and efficient algorithm including the treatment of singularities. The approach is illustrated by experiments on canopy models. This work [18] is going to be published.

#### Geometry of parametrized bicubic surfaces

Participants : André Galligo, Mike Stillman.

We start the study of the problem of describing the double point locus of a bicubic surface. Our motivation is to determine, whether a real bicubic patch over the unit square will have self-intersections. And if so, to identify useful points and curves in order to determine basic feature and to help analysis the surface accurately. The work was presented at the conference in honor of D. Lazard in Paris.

#### Classification of parameterised surfaces

Participants : Mohamed Elkadi, André Galligo, Thi-Ha Le.

Parametrized surfaces of low degrees are very useful in applications, specially in Computer Aided Geometric Design and Geometric Modeling. The precise description of their geometry is not easy in general. Here we study surfaces of bidegree (1, 2). We show that, generically up to linear changes of coordinates, they are classified by two continuous parameters (modulus). We present an elegant combinatorial description where these modulus appear as cross ratios. We provide compact implicit equations for these surfaces and for their singular locus together with a geometric interpretation (see [22]).

#### Semi-implicit representation of algebraic surfaces

Participants : Laurent Busé, André Galligo.

We continued our work on this new representation of algebraic surfaces which is an interesting intermediate representation between parameterized and implicit representations. In a work accepted for publication, we first develop further the theoretical side by giving a general definition in the language of projective complex algebraic geometry. Then we applied it to investigate the intersection of two bi-cubic surfaces, these surfaces are widely used in Computer Aided Geometric Design. In [20] we mainly addressed the topic of performing the usual CAD operations with semi-implicit representation of surfaces. We derived formulae for computing the normal and the curvatures at a regular point. We provided exact algorithms for computing self-intersections of a surface and more generally its singular locus. We also presented some surface/surface intersection algorithms relying on generalized resultant calculations.

#### Autointersection

Participants : André Galligo, Bernard Mourrain, Jean-Pascal Pavone.

In CAGD it is important not only to detect but also to describe and compute the self-intersection curves of surfaces bounding a solid constructed via usual CAGD operators. Such surfaces are called procedural: they are parametric but not necessarily rational. J.P. Pavone developed an efficient algorithm, based on sampling, to compute the self-intersection locus of such a parametric surface. It relies on the segmentation of the parameter domain into subdomains on which the parameterization map is injective, then applies adapted multiple surface/surface intersections. As the approach is sampling-based, there is no guarantee that all self-intersection points are located, also there is not guarantee that singular cases can be handled. Sampling based methods or lattice evaluation is used to a great extent within CAGD systems for surface-surface intersection calculation. For detecting self-intersections this approach is novel. The implementation of this algorithm has been connected to an industrial CAGD tool. Experimentations in this context shows the efficiency of the approach.

#### Arrangements of quadrics

Participants : Bernard Mourrain, Jean-Pierre Técourt, Monique Teillaud.

In the paper [17], we study a sweeping algorithm for computing the arrangement of a set of quadrics in . We define a ``trapezoidal'' decomposition in the sweeping plane, and we study the evolution of this subdivision during the sweep. A key point of this algorithm is the manipulation of algebraic numbers. In this perspective, we put a large emphasis on the use of algebraic tools, needed to compute the arrangement, including Sturm sequences and Rational Univariate Representation of the roots of a multivariate polynomial system.

#### Algebraic identities for the configurations of pairs of conics and quadrics

Participant : Emmanuel Briand.

Emmanuel Briand continued his work started at the *University of
Cantabria* in Santander on algebraic identities – equations, inequations,
inequalities – characterizing the configurations of pairs of projective
conics, and studied applications for arrangements of conics and
generalizations to quadrics. A publication is in preparation answering a
question about the characterization of the configurations of pairs of conics
by means of the signature function, raised in [56]. The
similar question about pairs of quadrics is also under study.

#### Topology of curves and surfaces

Participants : Grégory Gatellier, Georges Comte [UNSA], Bernard Mourrain, Jean-Pierre Técourt.

In this work, we can distinguish two parts, one concerning the topology of implicit curves (defined by two implicit equations) [24] to be published and another on algebraic surfaces (defined by one equation) which is still in progress. In both cases, starting from one or two implicit equations, the algorithm outputs an isotopic meshing of the curve or surface. Both algorithms are based on the strategy of sweeping algorithms: we choose a direction of sweeping, compute when the topology of the sections (with respect to the sweeping) change. We compute the topology of the sections and connect them. The main ingredients are projection tools, based on resultants and 0-dimensional solvers and from a more theoretical point of view, singularity theory with Whitney stratification.

#### Meshing real algebraic surfaces

Participants : Lionel Alberti, George Comte [UNSA], Bernard Mourrain.

We develop a new method to mesh surfaces defined by an algebraic equation, which is able to isolate the singular points of the surface, to guaranty the topology in the smooth part, and to give a topological model of singularity elsewhere, while producing a number of linear pieces, related to the Vitushkin variations of the surface. It applies to surfaces defined by a polynomial equation or a B-spline equation. We use Bernstein bases to represent the function in a box and subdivide this representation according to a generalization of Descartes rule, until the problem in each box boils down to the case where either the implicit object is proved to be homeomorph to its linear approximation in the cell or the size of the cell is smaller than . This ensures that the topology of the smooth part of an implicit surface is cached within a precision , where is a tunable parameter. For the singular cases, we use a combination of tools coming from singularity theory, real geometry and algebraic geometry. In particular, using the notion of Whitney stratification and Milnor balls, the method allows us to compute a finite partition of the space in cubes so that the zero set of the polynomial in each cube has the same topology as a cone. Once one has such a decomposition, it is easy to build a mesh for the zero set. The actual computation of the Whitney stratification is done using projective resultants. Part of this work is described in [29] and submitted to the proceedings of the conference ``Mathematical methods for curves and surfaces'' 2004, Tromsoe, Norway.

#### Applications to Computer Vision

Participants : Jean Ponce [University of Illinois at Urbana-Champaign], Theo Papadopoulo [Odyssée], Monique Teillaud, Bill Triggs [Lear].

In a work in progress, we introduce the *absolute quadratic
complex* formed by all lines that intersect the absolute conic. A
simple relation between a camera's intrinsic parameters, its
projection matrix expressed in a projective coordinate frame, and the
metric upgrade separating this frame from a metric one, provides a new
framework for autocalibration, particularly well suited to typical
digital cameras with rectangular or square pixels since the skew and
aspect ratio are decoupled from the other intrinsic parameters.