## Section: New Results

Keywords : computational geometry, robustness, perturbation, degeneracies, predicates, algebraic degree of algorithms.

### Geometric computing

#### Perturbations and vertex removal in Delaunay and regular 3D triangulations

Participants : Olivier Devillers, Monique Teillaud.

Though Delaunay and regular triangulations are very well known geometric data structures, the problem of the robust removal of a vertex in a three-dimensional triangulation is actually a problem in practice. We propose a simple method that allows to remove any vertex even when the points are in very degenerate configurations [61] . The solution is available in cgal (see Section 6.8.11 ).

#### Operations on curves

Participants : Pedro Machado Manhães de Castro, Sylvain Pion, Daniel Russel, Ilya Suslov, Monique Teillaud [ contact person ] , Elias Tsigaridas.

Work on the cgal curved kernel has continued, following the past years work, in two main directions.

The first direction consisted in research on methods to improve the new package for manipulations of circular arcs in the plane that was released in cgal 3.2 [71] (see Section 6.8.6 ). In particular, different possible representations of algebraic numbers were studied and compared. Geometric filtering techniques were also studied [74] , [70] , [69] . These improvements will be included in the next release. An extension of this work to the 3D case, which raises new issues, is in progress. This work is also mentioned in [27] .

The second research direction consists in defining concepts for the manipulation of algebraic curves of general degree, with the goal of a cgal implementation. This is done in collaboration with our ACS and Arcadia partners [52] , [51] .

#### Robust construction of the extended three-dimensional flow complex

Participants : Frédéric Cazals, Sylvain Pion.

The Delaunay triangulation and its dual, the Voronoi diagram, are ubiquitous geometric complexes. From a topological standpoint, the connexion has recently been made between these constructions and the Morse theory of distance functions. In particular, algorithms have been designed to compute the flow complex induced by the distance functions to a point set.

This paper [57] develops the first complete and robust construction of the extended flow complex, which in addition of the stable manifolds of the flow complex, also features the unstable manifolds. A first difficulty comes from the interplay between the degenerate cases of Delaunay and those which are flow specific. A second class of problems comes from cascaded constructions and predicates —as opposed to the standard in-circle and orientation predicates for Delaunay. We deal with both aspects and show how to implement a complete and robust flow operator, from which the extended flow complex is easily computed. We also present experimental results.

#### Lazy exact evaluation of geometric computations

Participant : Sylvain Pion.

In collaboration with A. Fabri (Geometry Factory )

We present [41] a generic C++ design to perform efficient and exact geometric computations using lazy evaluations. Exact geometric computations are critical for the robustness of geometric algorithms. Their efficiency is also critical for most applications, hence the need for delaying the exact computations at run time until they are actually needed. Our approach is generic and extensible in the sense that it is possible to make it a library that users can extend to their own geometric objects or primitives. It involves techniques such as generic functor adaptors, dynamic polymorphism, reference counting for the management of directed acyclic graphs and exception handling for detecting cases where exact computations are needed. It also relies on multiple precision arithmetic as well as interval arithmetic. We apply our approach to the whole geometric kernel of cgal .

#### On real root isolation

Participant : Elias Tsigaridas.

In collaboration with Ioannis Emiris (National University of Athens, Greece).

[62] presents the average-case bit complexity of subdivision-based univariate solvers, namely those named after Sturm, Descartes, and Bernstein. By real solving we mean real root isolation. We prove bounds of for all methods, where N bounds the polynomial degree and the coefficient bitsize, whereas their worst-case complexity is in . In the case of the Sturm solver, our bound depends on the number of real roots. Our work is a step towards understanding the practical complexity of real root isolation. This enables a better juxtaposition against numerical solvers, the latter having worst-case complexity in .

In [81] , we present algorithmic, complexity and implementation results concerning real root isolation of integer univariate polynomials using the continued fraction expansion of real algebraic numbers. One motivation is to explain the method's good performance in practice. We improve the previously known bound by a factor of d , where d is the polynomial degree and bounds the coefficient bit size, thus matching the current record complexity for real root isolation by exact methods (Sturm, Descartes and Bernstein subdivision). Namely our complexity bound is using a standard bound on the expected bit size of the integers in the continued fraction expansion. Moreover, using a homothetic transformation we improve the expected complexity bound to under the assumption that . We compute the multiplicities within the same complexity and extend the algorithm to non square-free polynomials.