Exact polynomial system solving for robust geometric computation

I describe an exact method for computing roots of a system of multivariate polynomials with rational coefficients, called the rational univariate reduction. This method enables performance of exact algebraic computation of coordinates of the roots of polynomials. In computational geometry, curves, surfaces and points are described as polynomials and their intersections. Thus, exact computation of the roots of polynomials allows the development and implementation of robust geometric algorithms. I describe applications in robust geometric modeling. In particular, I show a new method, called numerical perturbation scheme, that can be used successfully to detect and handle degenerate configurations appearing in boundary evaluation problems. I develop a derandomized version of the algorithm for computing the rational univariate reduction for a square system of multivariate polynomials and a new algorithm for a non-square system. I show how to perform exact computation over algebraic points obtained by the rational univariate reduction. I give a formal description of numerical perturbation scheme and its implementation

Restructuring Expression Dags for Efficient Parallelization

In the field of robust geometric computation it is often necessary to make exact decisions based on inexact floating-point arithmetic. One common approach is to store the computation history in an arithmetic expression dag and to re-evaluate the expression with increasing precision until an exact decision can be made. We show that exact-decisions number types based on expression dags can be evaluated faster in practice through parallelization on multiple cores. We compare the impact of several restructuring methods for the expression dag on its running time in a parallel environment

Robustness in geometric modeling - an intuitionistic and tolerance-based approach

Journal ArticleAn intuitionistic geometry approach is taken to develop two tolerance-based methods for robust geometric computation. The so called analytic model method and the approximated model method are developed independently of a specific application or a geometric algorithm. Geometric robustness is formally defined. Geometric relations are computed based on tolerances defined for geometric objects. Dynamic tolerance updating rules are given to preserve properties of the geometric relations. The two methods differ in the definition of robustness and they use different tolerance updating rules, and hence, they preseve different properties and are suitable for different kinds of applications. To handle the possibly occuring ambiguities dynamic ambiguity handling methods are described as well

Planar shape manipulation using approximate geometric primitives

We present robust algorithms for set operations and Euclidean transformations of curved shapes in the plane using approximate geometric primitives. We use a refinement algorithm to ensure consistency. Its computational complexity is \bigo(n\log n+k) for an input of size $n$ with k=\bigo(n^2) consistency violations. The output is as accurate as the geometric primitives. We validate our algorithms in floating point using sequences of six set operations and Euclidean transforms on shapes bounded by curves of algebraic degree~1 to~6. We test generic and degenerate inputs. Keywords: robust computational geometry, plane subdivisions, set operations

Counterexamples to the uniformity conjecture

AbstractThe Exact Geometric Computing approach requires a zero test for numbers which are built up using standard operations starting with the natural numbers. The uniformity conjecture, part of an attempt to solve this problem, postulates a simple linear relationship between the syntactic length of expressions built up from the natural numbers using field operations, radicals and exponentials and logarithms, and the smallness of non zero complex numbers defined by such expressions. It is shown in this article that this conjecture is incorrect, and a technique is given for generating counterexamples. The technique may be useful to check other conjectured constructive root bounds of this kind. A revised form of the uniformity conjecture is proposed which avoids all the known counterexamples

Summation Problem Revisited -- More Robust Computation

Numerical data processing is a key task across different fields of computer technology use. However, even simple summation of values is not precise due to the floating point representation use. This paper presents a practical algorithm for summation of values convenient for medium and large data sets. The proposed algorithm is simple, easy to implement. Its computational complexity is O(N) in the contrary of the Exact Sign Summation Algorithm (ESSA) approach with O(N^2) run-time complexity. The proposed algorithm is especially convenient for cases when exponent data differ significantly and many small values are summed with higher valuesComment: 9 pages, 3 Figs, 3 Tabs. Presented at Recent Advances in Computer Science Conf, 201

Robust boolean set operations for manifold solids bounded by planar and natural quadric surfaces

Journal ArticleThis paper describes our latest effort in robust solid modeling. An algorithm for set operations on solids bounded by planar and natural quadric surfaces, that handles all geometrically degenerate cases robustly, is described. We identify as the main reason for the lack of robustness in geometric modeling, that dependent relations are handled inconsistently by disregarding the dependencies. Instead of using explicit reasoning to make dependent decisions consistent, we show that redundant computation can be avoided by correctly ordering the operations, and redundant data can be eliminated in the set operation algorithm, so that the result is guaranteed to be a valid two-manifold solid