4,861 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
Faster Geometric Algorithms via Dynamic Determinant Computation
The computation of determinants or their signs is the core procedure in many
important geometric algorithms, such as convex hull, volume and point location.
As the dimension of the computation space grows, a higher percentage of the
total computation time is consumed by these computations. In this paper we
study the sequences of determinants that appear in geometric algorithms. The
computation of a single determinant is accelerated by using the information
from the previous computations in that sequence.
We propose two dynamic determinant algorithms with quadratic arithmetic
complexity when employed in convex hull and volume computations, and with
linear arithmetic complexity when used in point location problems. We implement
the proposed algorithms and perform an extensive experimental analysis. On one
hand, our analysis serves as a performance study of state-of-the-art
determinant algorithms and implementations. On the other hand, we demonstrate
the supremacy of our methods over state-of-the-art implementations of
determinant and geometric algorithms. Our experimental results include a 20 and
78 times speed-up in volume and point location computations in dimension 6 and
11 respectively.Comment: 29 pages, 8 figures, 3 table
Formally Certified Floating-Point Filters For Homogeneous Geometric Predicates
International audienceFloating-point arithmetic provides a fast but inexact way of computing geometric predicates. In order for these predicates to be exact, it is important to rule out all the numerical situations where floating-point computations could lead to wrong results. Taking into account all the potential problems is a tedious work to do by hand. We study in this paper a floating-point implementation of a filter for the orientation-2 predicate, and how a formal and partially automatized verification of this algorithm avoided many pitfalls. The presented method is not limited to this particular predicate, it can easily be used to produce correct semi-static floating-point filters for other geometric predicates
Formal certification of arithmetic filters for geometric predicates
International audienceFloating-point arithmetic provides a fast but inexact way of computing geometric predicates. In order for these predicates to be exact, it is important to rule out all the numerical situations where floating-point computations could lead to wrong results. Taking into account all the potential problems is a tedious work to do by hand. We study in this paper a floating-point implementation of a filter for the orientation-2 predicate, and how a formal and partially automatized verification of this algorithm avoided many pitfalls. The presented method is not limited to this particular predicate, it can easily be used to produce correct semi-static floating-point filters for other geometric predicates
A Probabilistic Analysis of the Power of Arithmetic Filters
The assumption of real-number arithmetic, which is at the basis of
conventional geometric algorithms, has been seriously challenged in recent
years, since digital computers do not exhibit such capability.
A geometric predicate usually consists of evaluating the sign of some
algebraic expression. In most cases, rounded computations yield a reliable
result, but sometimes rounded arithmetic introduces errors which may invalidate
the algorithms. The rounded arithmetic may produce an incorrect result only if
the exact absolute value of the algebraic expression is smaller than some
(small) varepsilon, which represents the largest error that may arise in the
evaluation of the expression. The threshold varepsilon depends on the structure
of the expression and on the adopted computer arithmetic, assuming that the
input operands are error-free.
A pair (arithmetic engine,threshold) is an "arithmetic filter". In this paper
we develop a general technique for assessing the efficacy of an arithmetic
filter. The analysis consists of evaluating both the threshold and the
probability of failure of the filter.
To exemplify the approach, under the assumption that the input points be
chosen randomly in a unit ball or unit cube with uniform density, we analyze
the two important predicates "which-side" and "insphere". We show that the
probability that the absolute values of the corresponding determinants be no
larger than some positive value V, with emphasis on small V, is Theta(V) for
the which-side predicate, while for the insphere predicate it is Theta(V^(2/3))
in dimension 1, O(sqrt(V)) in dimension 2, and O(sqrt(V) ln(1/V)) in higher
dimensions. Constants are small, and are given in the paper.Comment: 22 pages 7 figures Results for in sphere test inproved in
cs.CG/990702
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
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