196,947 research outputs found
Recent progress in exact geometric computation
AbstractComputational geometry has produced an impressive wealth of efficient algorithms. The robust implementation of these algorithms remains a major issue. Among the many proposed approaches for solving numerical non-robustness, Exact Geometric Computation (EGC) has emerged as one of the most successful. This survey describes recent progress in EGC research in three key areas: constructive zero bounds, approximate expression evaluation and numerical filters
Inner and Outer Rounding of Boolean Operations on Lattice Polygonal Regions
Robustness problems due to the substitution of the exact computation on real
numbers by the rounded floating point arithmetic are often an obstacle to
obtain practical implementation of geometric algorithms. If the adoption of the
--exact computation paradigm--[Yap et Dube] gives a satisfactory solution to
this kind of problems for purely combinatorial algorithms, this solution does
not allow to solve in practice the case of algorithms that cascade the
construction of new geometric objects. In this report, we consider the problem
of rounding the intersection of two polygonal regions onto the integer lattice
with inclusion properties. Namely, given two polygonal regions A and B having
their vertices on the integer lattice, the inner and outer rounding modes
construct two polygonal regions with integer vertices which respectively is
included and contains the true intersection. We also prove interesting results
on the Hausdorff distance, the size and the convexity of these polygonal
regions
On moduli of rings and quadrilaterals: algorithms and experiments
Moduli of rings and quadrilaterals are frequently applied in geometric
function theory, see e.g. the Handbook by K\"uhnau. Yet their exact values are
known only in a few special cases. Previously, the class of planar domains with
polygonal boundary has been studied by many authors from the point of view of
numerical computation. We present here a new -FEM algorithm for the
computation of moduli of rings and quadrilaterals and compare its accuracy and
performance with previously known methods such as the Schwarz-Christoffel
Toolbox of Driscoll and Trefethen. We also demonstrate that the -FEM
algorithm applies to the case of non-polygonal boundary and report results with
concrete error bounds
Intersection of paraboloids and application to Minkowski-type problems
In this article, we study the intersection (or union) of the convex hull of N
confocal paraboloids (or ellipsoids) of revolution. This study is motivated by
a Minkowski-type problem arising in geometric optics. We show that in each of
the four cases, the combinatorics is given by the intersection of a power
diagram with the unit sphere. We prove the complexity is O(N) for the
intersection of paraboloids and Omega(N^2) for the intersection and the union
of ellipsoids. We provide an algorithm to compute these intersections using the
exact geometric computation paradigm. This algorithm is optimal in the case of
the intersection of ellipsoids and is used to solve numerically the far-field
reflector problem
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
Evaluating Signs of Determinants Using Single-Precision Arithmetic
We propose a method of evaluating signs of 2×2 and 3×3 determinants with b-bit integer entries using only b and (b + 1)-bit arithmetic, respectively. This algorithm has numerous applications in geometric computation and provides a general and practical approach to robustness. The algorithm has been implemented and compared with other exact computation methods
Fast and exact computation of moments using discrete Green's theorem
Green's theorem evaluates a double integral over the region of an object by a simple integration along the boundary of the object. It has been used in moment computation since the shape of a binary object is totally determined by its boundary. By using a discrete analogue of Green's theorem, we present a new algorithm for fast computation of geometric moments. The algorithm is faster than previous methods, and gives exact results. The importance of exact computation is discussed by examining the invariance of Hu's moments. A fast method for computing moments of regions in grey level image, using discrete Green's theorem, is also presented
- …