21 research outputs found
A software framework for computing Newton polytopes of resultants and (reduced) discriminants
We present a new software for computing Newton polytopes
of resultant and discriminant polynomials. We illustrate its use with a number of examples
The maximum number of faces of the Minkowski sum of three convex polytopes
We derive tight expressions for the maximum
number of -faces, , of the
Minkowski sum, , of three -dimensional convex polytopes , and in ,
as a function of the number of vertices of the polytopes, for any .
Expressing the Minkowski sum as a section of the Cayley polytope of its summands, counting the -faces of reduces to counting the -faces of which meet the vertex sets of the three polytopes.
In two dimensions our expressions reduce to known results,
while in three dimensions, the tightness of our bounds follows by exploiting known tight bounds for the number of faces of -polytopes in , where .
For , the maximum values are attained when
, and are -polytopes, whose vertex sets are chosen appropriately from three distinct -dimensional moment-like curves
Sparse implicitization by interpolation: Characterizing non-exactness and an application to computing discriminants
We revisit implicitization by interpolation in order to examine its properties in the context of sparse elimination theory. Based on the computation of a superset of the implicit support, implicitization is reduced to computing the nullspace of a numeric matrix. The approach is applicable to polynomial and rational parameterizations of curves and (hyper)surfaces of any dimension, including the case of parameterizations with base points.
Our support prediction is based on sparse (or toric) resultant theory, in order to exploit the sparsity of the input and the output. Our method may yield a multiple of the implicit equation: we characterize and quantify this situation by relating the nullspace dimension to the predicted support and its geometry. In this case, we obtain more than one multiples of the implicit equation; the latter can be obtained via multivariate polynomial gcd (or factoring).
All of the above techniques extend to the case of approximate computation, thus yielding a method of sparse approximate implicitization, which is important in tackling larger problems. We discuss our publicly available Maple implementation through several examples, including the benchmark of bicubic surface.
For a novel application, we focus on computing the discriminant of a multivariate polynomial, which characterizes the existence of multiple roots and generalizes the resultant of a polynomial system.
This yields an efficient, output-sensitive algorithm for
computing the discriminant polynomial
An Output-sensitive Algorithm for Computing Projections of Resultant Polytopes
We develop an incremental algorithm to compute the Newton polytope
of the resultant, aka resultant polytope, or its
projection along a given direction.
The resultant is fundamental in algebraic elimination and
in implicitization of parametric hypersurfaces.
Our algorithm exactly computes vertex- and halfspace-representations
of the desired polytope using an oracle producing resultant vertices in a
given direction.
It is output-sensitive as it uses one oracle call per vertex.
We overcome the bottleneck of determinantal predicates
by hashing, thus accelerating execution from to times.
We implement our algorithm using the experimental CGAL package {\tt
triangulation}.
A variant of the algorithm computes successively tighter inner and outer
approximations: when these polytopes have, respectively,
90\% and 105\% of the true volume, runtime is reduced up to times.
Our method computes instances of -, - or -dimensional polytopes
with K, K or vertices, resp., within hr.
Compared to tropical geometry software, ours is faster up to
dimension or , and competitive in higher dimensions
Implicitization of curves and (hyper)surfaces using predicted support
We reduce implicitization of rational planar parametric curves and (hyper)surfaces to linear algebra, by interpolating the coefficients of the implicit equation.
For predicting the implicit support, we focus on methods that exploit input and output structure in the sense of sparse (or toric) elimination theory, namely by computing the Newton polytope of the implicit polynomial, via sparse resultant theory.
Our algorithm works even in the presence of base points but, in this case, the implicit equation shall be obtained as a factor of the produced polynomial.
We implement our methods on Maple, and some on Matlab as well, and study their numerical stability and efficiency on several classes of curves and surfaces.
We apply our approach to approximate implicitization,
and quantify the accuracy of the approximate output,
which turns out to be satisfactory on all tested examples; we also relate our measures to Hausdorff distance.
In building a square or rectangular matrix, an important issue is (over)sampling the given curve or surface: we conclude that unitary complexes offer the best tradeoff between speed and accuracy when numerical methods are employed, namely SVD, whereas for exact kernel computation random integers is the method of choice.
We compare our prototype to existing software and find that it is rather competitive
Sparse implicitization by interpolation: Geometric computations using matrix representations
Based on the computation of a superset of the implicit support,
implicitization of a parametrically given hyper-surface is reduced to computing
the nullspace of a numeric matrix. Our approach exploits the sparseness of the
given parametric equations and of the implicit polynomial. In this work, we
study how this interpolation matrix can be used to reduce some key geometric
predicates on the hyper-surface to simple numerical operations on the matrix,
namely membership and sidedness for given query points. We illustrate our
results with examples based on our Maple implementation
Interpolation of syzygies for implicit matrix representations
We examine matrix representations of curves and surfaces based on syzygies and constructed by interpolation through points. They are implicit representations of objects given as point clouds. The corresponding theory, including moving lines, curves and surfaces, has been developed for parametric models. Our contribution is to show how to compute the required syzygies by interpolation, when the geometric object is given by a point cloud whose sampling satisfies mild assumptions. We focus on planar and space curves, where the theory of syzygies allows us to design an exact algorithm yielding the optimal implicit expression. The method extends readily to surfaces without base points defined over triangular patches. Our Maple implementation has served to produce the examples in this paper and is available upon demand by the authors