Section: New Results

Non-linear computational geometry

Participants : Guillaume Moroz, Sylvain Lazard, Marc Pouget, Laurent Dupont, 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 (see Section 6.1 ), we work on the problem of solving a system of two bivariate polynomials. We are interested in certified numerical approximations or, more precisely, isolating boxes of the solutions. But we are also interested in computing, as intermediate symbolic objects, a Rational Univariate Representation (RUR) that is, roughly speaking, a univariate polynomial and two rational functions that map the roots of the univariate polynomial to the two coordinates of the solutions of the system. RURs are relevant symbolic objects because they allow to turn many queries on the system into queries on univariate polynomials. However, such representations require the computation of a separating form for the system, that is a linear combination of the variables that takes different values when evaluated at the distinct solutions of the system.

We published this year [11] results showing that, given two polynomials of degree at most $d$ with integer coefficients of bitsize at most $\tau$, (i) a separating form, (ii) the associated RUR, and (iii) isolating boxes of the solutions can be computed in, respectively, ${\tilde{O}}_B(d^8+d^7\tau )$, ${\tilde{O}}_B(d^7+d^6\tau )$ and ${\tilde{O}}_B(d^8+d^7\tau )$ bit operations in the worst case, where $\tilde{O}$ refers to the complexity where polylogarithmic factors are omitted and $O_B$ refers to the bit complexity.

However, during the publishing process, we have substentially improved these results. We have presented for these three sub-problems new algorithms that have worst-case bit complexity ${\tilde{O}}_B(d^6+d^5\tau )$. We have also presented probabilistic Las Vegas variants of our two first algorithms, which have expected bit complexity ${\tilde{O}}_B(d^5+d^4\tau )$. We also show that it is likely difficult to improve these complexities as it would essentially require to improve bounds on other fundamental problems (e.g., computing resultants, checking squarefreeness and root isolation of univariate polynomials) that have hold for decades.

This work was done in collaboration with Yacine Bouzidi (Inria Saclay), Michael Sagraloff (MPII Sarrebruken, Germany) and Fabrice Rouillier (Inria Rocquencourt). It is published in the research report [22] and submitted to a journal.

A key ingredient of the above work is the classical triangular decomposition algorithm via subresultants [31] on which we obtain two results of independent interest. First, we improved by a factor $d$ the state-of-the-art worst-case bit complexity of this algorithm [22] . One constraint on this algorithm is that it requires that the curves defined by the input polynomials have no common vertical asymptotes. Our second result is a generalization of this algorithm, which removes that restriction while preserving the same worst-case bit complexity of ${\tilde{O}}_B(d^6+d^5\tau )$. Furthermore, we actually present a refined bit complexity in ${\tilde{O}}_B(d_x^3{d}_y^3+(d_x^2{d}_y^3+d_x{d}_y^4)\tau )$ where $d_x$ and $d_y$ bound the degrees of the input polynomials in $x$ and $y$, respectively. We also prove that the total bitsize of the decomposition is in $\tilde{O}\left((d_x^2{d}_y^3+d_x{d}_y^4)\tau \right)$.

This work was done in collaboration with Fabrice Rouillier (Inria Rocquencourt). It is published in the research report [27] and submitted to a journal.

Numeric and Certified Isolation of the Singularities of the Projection of a Smooth Space Curve

Let a smooth real analytic curve embedded in ${ℝ}^3$ be defined as the solution of real analytic equations of the form $P(x,y,z)=Q(x,y,z)=0$ or $P(x,y,z)=\frac{\partial P}{\partial z}=0$. Our main objective is to describe its projection $𝒞$ onto the $(x,y)$-plane. In general, the curve $𝒞$ is not a regular submanifold of ${ℝ}^2$ and describing it requires to isolate the points of its singularity locus $\Sigma$. After describing the types of singularities that can arise under some assumptions on $P$ and $Q$, we present a new method to isolate the points of $\Sigma$. We experimented our method on pairs of independent random polynomials $(P,Q)$ and on pairs of random polynomials of the form $(P,\frac{\partial P}{\partial z})$ and got promising results [14] .

On the same topic but with a different approach, we improved our research report [26] by including experimental data using SubdivisionSolver (see Section 6.2 ) and submitted this work to a journal.

Mechanical design of parallel robots

In collaboration with F. Rouillier, D. Chablat and our PhD student Ranjan Jha, we analyzed the singularities and the workspace of different families of robots.

The first result is a certified description of the workspace and the singularities of a Delta like family robot [16] . Workspace and joint space analysis are essential steps in describing the task and designing the control loop of the robot, respectively. This paper presents the descriptive analysis of a family of delta-like parallel robots by using algebraic tools to induce an estimation about the complexity in representing the singularities in the workspace and the joint space. A Gröbner based elimination is used to compute the singularities of the manipulator and a Cylindrical Algebraic Decomposition algorithm is used to study the workspace and the joint space. From these algebraic objects, we propose some certified three dimensional plotting describing the shape of workspace and of the joint space which will help the engineers or researchers to decide the most suited configuration of the manipulator they should use for a given task. Also, the different parameters associated with the complexity of the serial and parallel singularities are tabulated, which further enhance the selection of the different configurations of the manipulator by comparing the complexity of the singularity equations.

The second result is an algebraic method to check the singularity-free paths for parallel robots [15] . Trajectory planning is a critical step while programming the parallel manipulators in a robotic cell. The main problem arises when there exists a singular configuration between the two poses of the end-effectors while discretizing the path with a classical approach. This paper presents an algebraic method to check the feasibility of any given trajectories in the workspace. The solutions of the polynomial equations associated with the trajectories are projected in the joint space using Gröbner based elimination methods and the remaining equations are expressed in a parametric form where the articular variables are functions of time $t$ unlike any numerical or discretization method. These formal computations allow to write the Jacobian of the manipulator as a function of time and to check if its determinant can vanish between two poses. Another benefit of this approach is to use a largest workspace with a more complex shape than a cube, cylinder or sphere. For the Orthoglide, a three degrees of freedom parallel robot, three different trajectories are used to illustrate this method.

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. We previously proposed a new algorithm for this problem, using a characterization of 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] , [28] . Unfortunately, our new algorithm is not yet efficient in practice. This year, we made several improvements on this algorithm. First, we decreased from 11 to 4 the degree of a critical polynomial that we need to solve and whose solutions induce the coefficients in some other polynomials appearing later in the computations. Second, we implemented Descarte's algorithm for isolating the real roots of univariate polynomials in the case where the coefficients belong to extensions of $\mathbb{Q}$ generated by at most two square roots. Furthermore, we are currently implementing the generalization of that algorithm when the coefficients belong to extensions of $\mathbb{Q}$ generated by one root of an arbitrary polynomial. These undergoing improvements should allow us to compute more directly the wanted reflexion point, thus avoiding problematic approximations and making the overall algorithm tractable.