62 research outputs found
Signature Sequence of Intersection Curve of Two Quadrics for Exact Morphological Classification
We present an efficient method for classifying the morphology of the
intersection curve of two quadrics (QSIC) in PR3, 3D real projective space;
here, the term morphology is used in a broad sense to mean the shape,
topological, and algebraic properties of a QSIC, including singularity,
reducibility, the number of connected components, and the degree of each
irreducible component, etc. There are in total 35 different QSIC morphologies
with non-degenerate quadric pencils. For each of these 35 QSIC morphologies,
through a detailed study of the eigenvalue curve and the index function jump we
establish a characterizing algebraic condition expressed in terms of the Segre
characteristics and the signature sequence of a quadric pencil. We show how to
compute a signature sequence with rational arithmetic so as to determine the
morphology of the intersection curve of any two given quadrics. Two immediate
applications of our results are the robust topological classification of QSIC
in computing B-rep surface representation in solid modeling and the derivation
of algebraic conditions for collision detection of quadric primitives
The Euclidean distance degree of an algebraic variety
The nearest point map of a real algebraic variety with respect to Euclidean
distance is an algebraic function. For instance, for varieties of low rank
matrices, the Eckart-Young Theorem states that this map is given by the
singular value decomposition. This article develops a theory of such nearest
point maps from the perspective of computational algebraic geometry. The
Euclidean distance degree of a variety is the number of critical points of the
squared distance to a generic point outside the variety. Focusing on varieties
seen in applications, we present numerous tools for exact computations.Comment: to appear in Foundations of Computational Mathematic
Computing the topology of a planar or space hyperelliptic curve
We present algorithms to compute the topology of 2D and 3D hyperelliptic
curves. The algorithms are based on the fact that 2D and 3D hyperelliptic
curves can be seen as the image of a planar curve (the Weierstrass form of the
curve), whose topology is easy to compute, under a birational mapping of the
plane or the space. We report on a {\tt Maple} implementation of these
algorithms, and present several examples. Complexity and certification issues
are also discussed.Comment: 34 pages, lot of figure
Likelihood Geometry
We study the critical points of monomial functions over an algebraic subset
of the probability simplex. The number of critical points on the Zariski
closure is a topological invariant of that embedded projective variety, known
as its maximum likelihood degree. We present an introduction to this theory and
its statistical motivations. Many favorite objects from combinatorial algebraic
geometry are featured: toric varieties, A-discriminants, hyperplane
arrangements, Grassmannians, and determinantal varieties. Several new results
are included, especially on the likelihood correspondence and its bidegree.
These notes were written for the second author's lectures at the CIME-CIRM
summer course on Combinatorial Algebraic Geometry at Levico Terme in June 2013.Comment: 45 pages; minor changes and addition
Intersection Testing between an Ellipsoid and an Algebraic Surface
International audienceThis paper presents a new method on the intersection testing problem between an ellipsoid and an algebraic surface. In the new method, the testing problem is turned into a new testing problem whether a univariate polynomial has a positive or negative real root. Examples are shown to illustrate the robustness and efficiency of the new method
- …