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

Algorithmic and topological aspects of semi-algebraic sets defined by quadratic polynomial

In this thesis, we consider semi-algebraic sets over a real closed field $R$ defined by quadratic polynomials. Semi-algebraic sets of $R^k$ are defined as the smallest family of sets in $R^k$ that contains the algebraic sets as well as the sets defined by polynomial inequalities, and which is also closed under the boolean operations (complementation, finite unions and finite intersections). We prove new bounds on the Betti numbers as well as on the number of different stable homotopy types of certain fibers of semi-algebraic sets over a real closed field $R$ defined by quadratic polynomials, in terms of the parameters of the system of polynomials defining them, which improve the known results. We conclude the thesis with presenting two new algorithms along with their implementations. The first algorithm computes the number of connected components and the first Betti number of a semi-algebraic set defined by compact objects in $\mathbb{R}^k$ which are simply connected. This algorithm improves the well-know method using a triangulation of the semi-algebraic set. Moreover, the algorithm has been efficiently implemented which was not possible before. The second algorithm computes efficiently the real intersection of three quadratic surfaces in $\mathbb{R}^3$ using a semi-numerical approach.Comment: PhD thesis, final version, 109 pages, 9 figure

On the asymptotic and practical complexity of solving bivariate systems over the reals

This paper is concerned with exact real solving of well-constrained, bivariate polynomial systems. The main problem is to isolate all common real roots in rational rectangles, and to determine their intersection multiplicities. We present three algorithms and analyze their asymptotic bit complexity, obtaining a bound of \sOB(N^{14}) for the purely projection-based method, and \sOB(N^{12}) for two subresultant-based methods: this notation ignores polylogarithmic factors, where $N$ bounds the degree and the bitsize of the polynomials. The previous record bound was \sOB(N^{14}). Our main tool is signed subresultant sequences. We exploit recent advances on the complexity of univariate root isolation, and extend them to sign evaluation of bivariate polynomials over two algebraic numbers, and real root counting for polynomials over an extension field. Our algorithms apply to the problem of simultaneous inequalities; they also compute the topology of real plane algebraic curves in \sOB(N^{12}), whereas the previous bound was \sOB(N^{14}). All algorithms have been implemented in MAPLE, in conjunction with numeric filtering. We compare them against FGB/RS, system solvers from SYNAPS, and MAPLE libraries INSULATE and TOP, which compute curve topology. Our software is among the most robust, and its runtimes are comparable, or within a small constant factor, with respect to the C/C++ libraries. Key words: real solving, polynomial systems, complexity, MAPLE softwareComment: 17 pages, 4 algorithms, 1 table, and 1 figure with 2 sub-figure

Hierarchical shape-based surface reconstruction for dense multi-view stereo

International audienceThe recent widespread availability of urban imagery has lead to a growing demand for automatic modeling from multiple images. However, modern image-based modeling research has focused either on highly detailed reconstructions of mostly small objects or on human-assisted simplified modeling. This paper presents a novel algorithm which automatically outputs a simplified, segmented model of a scene from a set of calibrated input images, capturing its essential geometric features. Our approach combines three successive steps. First, a dense point cloud is created from sparse depth maps computed from the input images. Then, shapes are robustly extracted from this set of points. Finally, a compact model of the scene is built from a spatial subdivision induced by these structures: this model is a global minimum of an energy accounting for the visibility of the final surface. The effectiveness of our method is demonstrated through several results on both synthetic and real data sets, illustrating the various benefits of our algorithm, its robustness and its relevance for architectural scenes

Complete Subdivision Algorithms, II: Isotopic Meshing of Singular Algebraic Curves

Given a real valued function f(X,Y), a box region B_0 in R^2 and a positive epsilon, we want to compute an epsilon-isotopic polygonal approximation to the restriction of the curve S=f^{-1}(0)={p in R^2: f(p)=0} to B_0. We focus on subdivision algorithms because of their adaptive complexity and ease of implementation. Plantinga and Vegter gave a numerical subdivision algorithm that is exact when the curve S is bounded and non-singular. They used a computational model that relied only on function evaluation and interval arithmetic. We generalize their algorithm to any bounded (but possibly non-simply connected) region that does not contain singularities of S. With this generalization as a subroutine, we provide a method to detect isolated algebraic singularities and their branching degree. This appears to be the first complete purely numerical method to compute isotopic approximations of algebraic curves with isolated singularities

Computation and Physics in Algebraic Geometry

Physics provides new, tantalizing problems that we solve by developing and implementing innovative and effective geometric tools in nonlinear algebra. The techniques we employ also rely on numerical and symbolic computations performed with computer algebra. First, we study solutions to the Kadomtsev-Petviashvili equation that arise from singular curves. The Kadomtsev-Petviashvili equation is a partial differential equation describing nonlinear wave motion whose solutions can be built from an algebraic curve. Such a surprising connection established by Krichever and Shiota also led to an entirely new point of view on a classical problem in algebraic geometry known as the Schottky problem. To explore the connection with curves with at worst nodal singularities, we define the Hirota variety, which parameterizes KP solutions arising from such curves. Studying the geometry of the Hirota variety provides a new approach to the Schottky problem. We investigate it for irreducible rational nodal curves, giving a partial solution to the weak Schottky problem in this case. Second, we formulate questions from scattering amplitudes in a broader context using very affine varieties and D-module theory. The interplay between geometry and combinatorics in particle physics indeed suggests an underlying, coherent mathematical structure behind the study of particle interactions. In this thesis, we gain a better understanding of mathematical objects, such as moduli spaces of point configurations and generalized Euler integrals, for which particle physics provides concrete, non-trivial examples, and we prove some conjectures stated in the physics literature. Finally, we study linear spaces of symmetric matrices, addressing questions motivated by algebraic statistics, optimization, and enumerative geometry. This includes giving explicit formulas for the maximum likelihood degree and studying tangency problems for quadric surfaces in projective space from the point of view of real algebraic geometry

Degree-Driven Design of Geometric Algorithms for Point Location, Proximity, and Volume Calculation

Correct implementation of published geometric algorithms is surprisingly difficult. Geometric algorithms are often designed for Real-RAM, a computational model that provides arbitrary precision arithmetic operations at unit cost. Actual commodity hardware provides only finite precision and may result in arithmetic errors. While the errors may seem small, if ignored, they may cause incorrect branching, which may cause an implementation to reach an undefined state, produce erroneous output, or crash. In 1999 Liotta, Preparata and Tamassia proposed that in addition to considering the resources of time and space, an algorithm designer should also consider the arithmetic precision necessary to guarantee a correct implementation. They called this design technique degree-driven algorithm design. Designers who consider the time, space, and precision for a problem up-front arrive at new solutions, gain further insight, and find simpler representations. In this thesis, I show that degree-driven design supports the development of new and robust geometric algorithms. I demonstrate this claim via several new algorithms. For n point sites on a UxU grid I consider three problems. First, I show how to compute the nearest neighbor transform in O(U^2) expected time, O(U^2) space, and double precision. Second, I show how to create a data structure in O(n log Un) expected time, O(n) expected space, and triple precision that supports O(log n) time and double precision post-office queries. Third, I show how to compute the Gabriel graph in O(n^2) time, O(n^2) space and double precision. For computing volumes of CSG models, I describe a framework that uses a minimal set of predicates that use at most five-fold precision. The framework is over 500x faster and two orders of magnitude more accurate than a Monte Carlo volume calculation algorithm.Doctor of Philosoph

Minkowski Sum Construction and other Applications of Arrangements of Geodesic Arcs on the Sphere

We present two exact implementations of efficient output-sensitive algorithms that compute Minkowski sums of two convex polyhedra in 3D. We do not assume general position. Namely, we handle degenerate input, and produce exact results. We provide a tight bound on the exact maximum complexity of Minkowski sums of polytopes in 3D in terms of the number of facets of the summand polytopes. The algorithms employ variants of a data structure that represents arrangements embedded on two-dimensional parametric surfaces in 3D, and they make use of many operations applied to arrangements in these representations. We have developed software components that support the arrangement data-structure variants and the operations applied to them. These software components are generic, as they can be instantiated with any number type. However, our algorithms require only (exact) rational arithmetic. These software components together with exact rational-arithmetic enable a robust, efficient, and elegant implementation of the Minkowski-sum constructions and the related applications. These software components are provided through a package of the Computational Geometry Algorithm Library (CGAL) called Arrangement_on_surface_2. We also present exact implementations of other applications that exploit arrangements of arcs of great circles embedded on the sphere. We use them as basic blocks in an exact implementation of an efficient algorithm that partitions an assembly of polyhedra in 3D with two hands using infinite translations. This application distinctly shows the importance of exact computation, as imprecise computation might result with dismissal of valid partitioning-motions.Comment: A Ph.D. thesis carried out at the Tel-Aviv university. 134 pages long. The advisor was Prof. Dan Halperi

Large bichromatic point sets admit empty monochromatic 4-gons

We consider a variation of a problem stated by Erd˝os and Szekeres in 1935 about the existence of a number fES(k) such that any set S of at least fES(k) points in general position in the plane has a subset of k points that are the vertices of a convex k-gon. In our setting the points of S are colored, and we say that a (not necessarily convex) spanned polygon is monochromatic if all its vertices have the same color. Moreover, a polygon is called empty if it does not contain any points of S in its interior. We show that any bichromatic set of n ≥ 5044 points in R2 in general position determines at least one empty, monochromatic quadrilateral (and thus linearly many).Postprint (published version