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 where
and denote respectively the maximum degree and bitsize of the input
(and where refers to the complexity where polylogarithmic
factors are omitted and refers to the bit complexity). This algorithm
simplifies and decreases by a factor 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 . 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 and a bound on the
bitsize of its coefficients in . We showed in
addition that isolating boxes of the solutions of the system can be computed
from the RUR with bit operations. Finally, we
showed how a RUR can be used to evaluate the sign of a bivariate polynomial (of
degree at most and bitsize at most ) at one real solution of the
system in bit operations and at all the
real solutions in only 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 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
(resp. ),
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 , while
ensuring that does not contain any closed loop of . We solve
this problem by providing a square deflation system that can be used to certify
numerically whether contains a singularity . Then we introduce a numeric
adaptive separation criterion based on interval arithmetic to ensure that the
topology of in is homeomorphic to the local topology at .
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 and going through another 3D point , 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 and . 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] .