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Section: New Results

Non-linear computational geometry

Participants : Guillaume Moroz, Sylvain Lazard, Marc Pouget, Mohamed Yacine Bouzidi, Laurent Dupont, Olive Chakraborty, Rémi Imbach.

Solving bivariate systems and topology of plane algebraic curves

In the context of our algorithm Isotop for computing the topology of plane algebraic curves [3] , we work on the problem of solving a system of two bivariate polynomials. We focus on the problem of computing a Rational Univariate Representation (RUR) of the solutions, that is, roughly speaking, a univariate polynomial and two rational functions which map the roots of the polynomial to the two coordinates of the solutions of the system. The PhD thesis of Yacine Bouzidi [10] presented several results on this theme obtained during the past three years.

Separating linear forms.    We addressed the problem of computing a linear separating form of a system of two bivariate polynomials with integer coefficients, that is a linear combination of the variables that takes different values when evaluated at the distinct solutions of the system. The computation of such linear forms is at the core of most algorithms that solve algebraic systems by computing rational parameterizations of the solutions and this is the bottleneck of these algorithms in terms of worst-case bit complexity. We presented for this problem a new algorithm of worst-case bit complexity ${\tilde{O}}_B(d^7+d^6\tau )$ where $d$ and $\tau$ denote respectively the maximum degree and bitsize of the input (and where $\tilde{O}$ refers to the complexity where polylogarithmic factors are omitted and $O_B$ refers to the bit complexity). This algorithm simplifies and decreases by a factor $d$ the worst-case bit complexity of a previous algorithm we presented in 2013 [24] . Our new algorithm also yields, for this problem, a probabilistic Las-Vegas algorithm of expected bit complexity ${\tilde{O}}_B(d^5+d^4\tau )$. These results were presented at the International Symposium on Symbolic and Algebraic Computation in 2014 [15] .

Solving bivariate systems & RURs.   Given such a separating linear form, we presented an algorithm for computing a RUR with worst-case bit complexity in ${\tilde{O}}_B(d^7+d^6\tau )$ and a bound on the bitsize of its coefficients in $\tilde{O}(d^2+d\tau )$. We showed in addition that isolating boxes of the solutions of the system can be computed from the RUR with ${\tilde{O}}_B(d^8+d^7\tau )$ bit operations. Finally, we showed how a RUR can be used to evaluate the sign of a bivariate polynomial (of degree at most $d$ and bitsize at most $\tau$) at one real solution of the system in ${\tilde{O}}_B(d^8+d^7\tau )$ bit operations and at all the $\Theta \left(d^2\right)$ real solutions in only $O\left(d\right)$ times that for one solution. These results were submitted in 2013, revised in 2014 and will appear in 2015 in the Journal of Symbolic Computation [12] .

This work is done in collaboration with Fabrice Rouillier (project-team Ouragan at Inria Paris-Rocquencourt).

Topology of the projection of a space curve

Let $𝒞$ be a real plane algebraic curve defined by the resultant of two polynomials (resp. by the discriminant of a polynomial). Geometrically, such a curve is the projection of the intersection of the surfaces $P(x,y,z)=Q(x,y,z)=0$ (resp. $P(x,y,z)=\frac{\partial P}{\partial z}(x,y,z)=0$), and generically its singularities are nodes (resp. nodes and ordinary cusps). State-of-the-art numerical algorithms cannot handle, in practice, the computation of the curve topology in non-trivial instances. The main challenge is to find numerical criteria that guarantee the existence and the uniqueness of a singularity inside a given box $B$, while ensuring that $B$ does not contain any closed loop of $𝒞$. We solve this problem by providing a square deflation system that can be used to certify numerically whether $B$ contains a singularity $p$. Then we introduce a numeric adaptive separation criterion based on interval arithmetic to ensure that the topology of $𝒞$ in $B$ is homeomorphic to the local topology at $p$. The theoretical parts of these results are summarized in [18] and are to be combined with experimental data before submission to a journal.

Reflection through quadric mirror surfaces

We addressed the problem of finding the reflection point on quadric mirror surfaces, especially ellipsoid, paraboloid or hyperboloid of two sheets, of a light ray emanating from a 3D point source $P_1$ and going through another 3D point $P_2$, the camera center of projection. This is a classical problem known as Alhazen's problem dating from around 1000 A.D. and based on the work of Ptolemy around 150 A.D. [22] , [27] . We proposed a new algorithm for this problem, using a characterization the reflection point as the tangential intersection point between the mirror and an ellipsoid with foci $P_1$ and $P_2$. The computation of this tangential intersection point is based on our algorithm for the computation of the intersection of quadrics [5] , [21] . The implementation is in progress. This work is done in collaboration with Nuno Gonçalves, University of Coimbra (Portugal).

Describing the workspace of a manipulator

We studied the geometry of the solutions of the 3-RPS parallel manipulator. In particular, a parallel manipulator usually has several solutions to the Direct Kinematic Problem. These solutions correspond to different assembly modes. A challenge is to find non-singular trajectories connecting different assembly modes. In the literature, this is done by encircling locally a cusp point of the discriminant variety in the joint space. In this work, we used tools from computer algebra to compute a partition of the work space in uniqueness domains. This allowed us to find global singularity-free trajectories reaching up to three assembly modes [16] , [17] .