Section: New Results

Non-linear Computational Geometry

Participants : Laurent Dupont, Rémi Imbach, Sylvain Lazard, Guillaume Moroz, Marc Pouget.

Numeric and Certified Algorithm for the Topology 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 $\Sigma$.

In previous work, we have shown how to describe the set of singularities $\Sigma$ of $𝒞$ as regular solutions of a so-called ball system suitable for a numerical subdivision solver. In our current work, the space curve is first enclosed in a set of boxes with a certified path-tracker to restrict the domain where the ball system is solved. Boxes around singular points are then computed such that the correct topology of the curve inside these boxes can be deduced from the intersections of the curve with their boundaries. The tracking of the space curve is then used to connect the smooth branches to the singular points. The subdivision of the plane induced by the curve is encoded as an extended planar combinatorial map allowing point location. This work is not already published but has been presented by R. Imbach at the Summer Workshop on Interval Methods (https://swim2016.sciencesconf.org/).

The technical report [28] describes the software SubdivisionSolver (see Section 5.2) used within this project.

A Fast Algorithm for Computing the Truncated Resultant

Let $P$ and $Q$ be two polynomials in $𝕂[x,y]$ with degree at most $d$, where $𝕂$ is a field. Denoting by $R\in 𝕂\left[x\right]$ the resultant of $P$ and $Q$ with respect to $y$, we present an algorithm to compute $Rmod{x}^k$ in $\tilde{O}\left(kd\right)$ arithmetic operations in $𝕂$, where the $\tilde{O}$ notation indicates that we omit polylogarithmic factors. This is an improvement over state-of-the-art algorithms that require to compute $R$ in $\tilde{O}\left(d^3\right)$ operations before computing its first $k$ coefficients [24].

This work was done in collaboration with Éric Schost (Waterloo University, Canda).

Quadric Arrangement in Classifying Rigid Motions of a 3D Digital Image

Rigid motions are fundamental operations in image processing. While bijective and isometric in ${ℝ}^3$, they lose these properties when digitized in ${\mathbb{Z}}^3$. To understand how the digitization of 3D rigid motions affects the topology and geometry of a chosen image patch, we classify the rigid motions according to their effect on the image patch. This classification can be described by an arrangement of hypersurfaces in the parameter space of 3D rigid motions of dimension six. However, its high dimensionality and the existence of degenerate cases make a direct application of classical techniques, such as cylindrical algebraic decomposition or critical point method, difficult. We show that this problem can be first reduced to computing sample points in an arrangement of quadrics in the 3D parameter space of rotations. Then we recover information about the three remaining parameters of translation. We implemented an ad-hoc variant of state-of-the-art algorithms and applied it to an image patch of cardinality 7. This leads to an arrangement of 81 quadrics and we recovered the classification in less than one hour on a machine equipped with 40 cores [25].

This work was done in collaboration with Kacper Pluta (LIGM - Laboratoire d'Informatique Gaspard-Monge), Yukiko Kenmochi (LIGM - Laboratoire d'Informatique Gaspard-Monge), Pascal Romon (LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées).

Influence of the Trajectory Planning on the Accuracy of the Orthoglide 5-axis manipulator

Usually, the accuracy of parallel manipulators depends on the architecture of the robot, the design parameters, the trajectory planning and the location of the path in the workspace. This paper reports the influence of static and dynamic parameters in computing the error in the pose associated with the trajectory planning made and analyzed with the Orthoglide 5-axis (Figure 1). An error model is proposed based on the joint parameters (velocity and acceleration) and experimental data coming from the Orthoglide 5-axis. Newton and Gröbner based elimination methods are used to project the joint error in the workspace to check the accuracy/error in the Cartesian space. For the analysis, five similar trajectories with different locations inside the workspace are defined using fifth order polynomial equation for the trajectory planning. It is shown that the accuracy of the robot depends on the location of the path as well as the starting and the ending posture of the manipulator due to the acceleration parameters [23].

This work was done in collaboration with Ranjan Jha (IRCCyN - Institut de Recherche en Communications et en Cybernétique de Nantes), Damien Chablat (IRCCyN - Institut de Recherche en Communications et en Cybernétique de Nantes), Fabrice Rouillier (Inria).

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 5.1), 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 the transformation of 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, ${\tilde{O}}_B(d^5+d^5\tau )$, 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. Furthermore, we also presented probabilistic Las Vegas variants of problems (i) and (ii), which have expected bit complexity ${\tilde{O}}_B(d^5+d^4\tau )$. We also showed that these complexities are “morally” optimal in the sense of that improving them would essentially require to improve bounds on several other fundamental problems (on resultants and roots isolation of univariate polynomials) that have hold for decades. These progresses are substential since, when we started woriking on these problems, their best know complexities were in ${\tilde{O}}_B(d^{12}+d^{10}{\tau }^2)$ (2009).

This work was done in collaboration with Yacine Bouzidi (Inria Lille), Michael Sagraloff (MPII Sarrebruken, Germany) and Fabrice Rouillier (Inria Rocquencourt).

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], [32]. 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 Descartes' 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. We are also interested by extensions decomposable in extensions of degree 2. These undergoing improvements should allow us to compute more directly the wanted reflection point, thus avoiding problematic approximations and making the overall algorithm tractable.