# Software of the team

## CGAL

We are participating to CGAL the computational geometry algorithms library.

## QI

QI stands for Quadrics Intersection. QI is the first exact, robust and efficient implementation of an algorithm for parameterizing the intersection of two arbitrary quadrics given in implicit form. QI computes an exact parameterization of the intersection of two quadrics with integer coefficients of arbitrary size with the following features:

- QI identifies, separates and parameterizes all the algebraic components (lines, conics, cubic, quartic) of the intersection;
- the parameterization is rational when one exists; otherwise the intersection is a non-singular quartic and the parameterization involves the square root of a polynomial;
- the coefficients of parameterization are small in the sense that the degree of the field extension (over the rationals) on which they are defined is minimal except in a small number of cases in which they may involve one extra possibly unnecessary square root;
- QI is fast and efficient and can routinely compute parameterizations of the intersection of quadrics with input coefficients having ten digits in less than 50 milliseconds on a mainstream PC.

## ISOTOP

Isotop is a Maple software for computing the topology/isotopy of an implicit algebraic plane curve, that is, for computing an arrangement of polylines isotopic to the input curve. It also provides a new solution to the important problem of plotting such curves in a certified way, that is -- in particular -- without missing connected components, singular points, etc. This problem is also a necessary key step for computing arrangements of algebraic curves.

TBD