Section: New Results

Certified geometric computing for curves and surfaces

Participants : Jinsan Cheng, Laurent Dupont, Hazel Everett, Sylvain Lazard, Luis Peñaranda, Maria Pentcheva, Sylvain Petitjean, Marc Pouget, George Tzoumas.

Voronoi diagram of polyhedra in 3D

We are working on the problem of computing the medial axis or Voronoi diagram of polyhedra in 3D. These structures are largely used in applications; the medial axis is, for instance, a way of representing a shape by its topological skeleton. Such a diagram is a partition of space into cells, each of which consists of the points closest to one particular object than to any other. Moreover, the set of points equidistant to two lines (or to a line and a point) is a quadric and the set of points equidistant to three lines is the intersection of two quadrics. While such structures are well-understood in the plane and for simple situations in higher dimensions (e.g. for sets of points), a lot remains to be done; for example, there is no working solution for computing exactly the medial axis of a polyhedron.

We started a few years ago by considering the Voronoi diagram of lines and we finally published this year, in the journal Discrete and Computational Geometry, some very nice results characterizing the topology of the Voronoi diagrams of three lines [15] . We proved, in particular, that the topology is invariant for lines in general position and we obtained a monotonicity property on the arcs of the diagram which has important algorithmic implications. The proof technique, which relies heavily upon modern tools of computer algebra, is also of great interest in its own right.

We have worked during the last two years on the problem of extending these results to the case of three lines in arbitrary positions providing the first complete characterization of the Voronoi diagram of any three lines. Preliminary results were presented this year at the European Workshop on Computational Geometry [25] . These results also yielded a new algorithm, fundamental for handling robustness issues, for sorting points along the arcs of Voronoi diagrams of lines (with rational coefficients) using only rational linear tests.

Adjacency graph of an arrangement of integer quadrics

We have presented, a couple years ago, a complete, exact and efficient algorithm and its implementation for computing the adjacency graph of an arrangement of quadrics with integer coefficients  [54] . This year, we completed this work and submitted it to the Journal of Symbolic Computation  [34] . This algorithm builds upon our previous work on parameterization of intersections of quadrics. Intersecting a parameterized intersection of two quadrics with a third quadric leads to finding the real zeros of polynomials of degree at most 8 with possibly algebraic coefficients. Experiments show that our implementation outperforms past approaches when dealing with generic situations, even in case of large bitsize and/or algebraic coefficients. This efficiency, even over algebraic extensions and with large bit-size numbers, is due, during the computation of the roots of univariate polynomials, to the use of the bitstream Descartes algorithm, which replaces each number by a series of certified approximations. In non-generic situations, the current implementation is hampered by slow gcd computations over algebraic extensions.

Algebraic tools for geometric computing

In computational geometry, many problems lead to standard, though difficult, algebraic questions such as computing the real roots of a system of equations, computing the sign of a polynomial at the roots of a system, or determining the dimension of a set of solutions. Our goal is two-fold. First, we want to make state-of-the-art algebraic software more accessible to the computational geometry community, in particular, through the computational geometric library CGAL. Second, our goal is to demonstrate to which extent such state-of-the-art certified algebraic root-finding systems can be used in geometric algorithms to obtain certified constructions involving curved objects without hindering performance. We have presented some results in these two directions at the 8th International Symposium on Experimental Algorithms, SEA'09 [22] , and we submitted this year, to the CGAL editorial board, a package which is a model of the Univariate Algebraic Kernel concept for algebraic computations (see section Software).

Topology and geometry of algebraic curves: Isotop

We worked over the last years on the problem of computing, in a certified way, the topology of algebraic curves, that is, computing an arrangement of polylines isotopic to the input curve. The objective here is to compute efficiently and in a certified way arrangements of algebraic curves. A necessary key step is to compute the topology of any given curve. Moreover, geometric information, such as the position of singular and critical points, is also mandatory for computing arrangements of several curves using a sweep-line algorithm. A difficulty is to compute efficiently this information for the given coordinate system even if the curve is not in generic position; previous practical approaches shear back and forth the coordinate system, which is time consuming. In addition, costly computations with polynomials whose coefficients are algebraic should be avoided. We have recently presented an algorithm that incorporates several improvements over previous methods and overcome these difficulties. In particular, our approach does not require generic position nor shearing. This work has been presented this year at the 25th annual Symposium on Computational Geometry [20] , and was submitted to the journal of Mathematics in Computer Science  [31] . We have also developed a Maple implementation of this algorithm which is very promising (see section Software).

Constant-complexity geometric problems and algebraic invariants

We have continued revisiting some key constant-complexity geometric problems with a view to better understand their degenerate instances and the geometric predicates underlying their detection. For that, we mostly rely on classical tools, in particular (classical) algebraic invariant century, which was perceived as a bridge between geometry and algebra by the mathematicians of the 19th century (culminating with Klein's Erlangen Program, and the view of geometry as the study of the properties of a space invariant under the action of a group of transformations). Last year, we studied the relative position of two plane projective conics and showed that it can be characterized by predicates of bidegree at most (6, 6) in the coefficients of the input conics  [63] , improving upon previous results. By relative position we mean the morphology of the intersection, the rigid isotopy class and which conic is inside the other when applicable. Analyzing the algebraic invariant theory of pencils of conics, we constructed a special conic – called a combinant – invariantly attached to a given pencil and showed how the projective type of this combinant, encoded by its inertia, is characteristic of the intersection type of the two conics in most cases. However, the problem was treated purely algebraically and the results have no obvious geometric meaning: why such inertia is characteristic of such intersection pattern is obscure. This year, we reproved essentially the same results, but using an entirely new approach which has the benefit of making perfectly clear the geometric meaning of the inertia of the combinant conic and overall bringing substantial geometric insight to the problem [23] . The key intermediate tool we use that illuminates this interpretation is the Bezoutian. We are working on extending these results to other primitives.

Bounded-curvature path planning

We studied the problem of computing shortest paths of bounded curvature that visits a sequence of n points in order. This problem, which has been open for about 15 years, is crucial for path planning of car-like robots in the presence of polygonal obstacles. We proved that, under some conditions, this problem reduces to optimizing a convex function over a convex n -dimensional domain. This result reveals a fundamental property of curvature-constrained paths among polygonal obstacles, and it provides the first efficient solution for this long-standing problem. This result has been submitted to the Symposium on Computational Geometry in December 2009 [36] .

Embedding geometric structures

This year, we started work on the problem of embedding geometric objects on a grid of . Essentially all industrial applications take, as input, models defined with a fixed-precision floating-point arithmetic, typically doubles. As a consequence, geometric objects constructed using exact arithmetic must be embedded on a fixed-precision grid before they can be used as input in other software. More precisely, the problem is, given a geometric object, to find a similar object representable with fixed-precision floating-point arithmetic, where similar means topologically equivalent, close according to some distance function, etc. We started working on the problem of rounding polyhedral subdivisions on a grid of , where the only known method, due to Fortune, induces a blow-up in the complexity that is inacceptable in practice. We worked so far on the simpler problem of embedding convex polyhedra. We also showed negative results that, even in the plane, convex polygons cannot be rounded while conserving both convexity and proximity of the rounded vertices. This project is joint work with Mark de Berg (Eindhoven University), Dan Halperin (Tel Aviv University) and Olivier Devillers (Geometrica, INRIA).