1,829 research outputs found
Classroom Examples of Robustness Problems in Geometric Computations
International audienceThe algorithms of computational geometry are designed for a machine model with exact real arithmetic. Substituting floating point arithmetic for the assumed real arithmetic may cause implementations to fail. Although this is well known, there is no comprehensive documentation of what can go wrong and why. In this extended abstract, we study a simple incremental algorithm for planar convex hulls and give examples which make the algorithm fail in all possible ways. We also show how to construct failure-examples semi-systematically and discuss the geometry of the floating point implementation of the orientation predicate. We hope that our work will be useful for teaching computational geometry. The full paper is available at http://hal.inria.fr/inria-00344310/. It contains further examples, more theory, and color pictures. We strongly recommend to read the full paper instead of this extended abstract
A programme to determine the exact interior of any connected digital picture
Region filling is one of the most important and fundamental operations in
computer graphics and image processing. Many filling algorithms and their
implementations are based on the Euclidean geometry, which are then translated
into computational models moving carelessly from the continuous to the finite
discrete space of the computer. The consequences of this approach is that most
implementations fail when tested for challenging degenerate and nearly
degenerate regions. We present a correct integer-only procedure that works for
all connected digital pictures. It finds all possible interior points, which
are then displayed and stored in a locating matrix. Namely, we present a
filling and locating procedure that can be used in computer graphics and image
processing applications
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
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
Geodesics in Heat
We introduce the heat method for computing the shortest geodesic distance to
a specified subset (e.g., point or curve) of a given domain. The heat method is
robust, efficient, and simple to implement since it is based on solving a pair
of standard linear elliptic problems. The method represents a significant
breakthrough in the practical computation of distance on a wide variety of
geometric domains, since the resulting linear systems can be prefactored once
and subsequently solved in near-linear time. In practice, distance can be
updated via the heat method an order of magnitude faster than with
state-of-the-art methods while maintaining a comparable level of accuracy. We
provide numerical evidence that the method converges to the exact geodesic
distance in the limit of refinement; we also explore smoothed approximations of
distance suitable for applications where more regularity is required
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