Finite-resolution computational geometry

@article{Greene1986FiniteresolutionCG,
  title={Finite-resolution computational geometry},
  author={Daniel H. Greene and F. Frances Yao},
  journal={27th Annual Symposium on Foundations of Computer Science (sfcs 1986)},
  year={1986},
  pages={143-152}
}
  • D. Greene, F. F. Yao
  • Published 27 October 1986
  • Mathematics, Computer Science
  • 27th Annual Symposium on Foundations of Computer Science (sfcs 1986)
Geometric algorithms are usually designed with continuous parameters in mind. When the underlying geometric space is intrinsically discrete, as is the case for computer graphics problems, such algorithms are apt to give invalid solutions if properties of a finite-resolution space are not taken into account. In this paper we discuss an approach for transforming geometric concepts and algorithms from the continuous domain to the discrete domain. As an example we consider the discrete version of… 
View on IEEE
researchgate.net
184 Citations
Highly Influential Citations
12
Background Citations
43
Methods Citations
24

184 Citations

Citation Type

Citation Type

Computational geometry on the grid: Transversal and planesweep algorithms for spatial applications
TLDR
This paper presents geometric algorithms that satisfy the properties required above and that rest on a discrete geometric basis in form of a uniform integer grid.
Recipes for geometry and numerical analysis - Part I: an empirical study
TLDR
This paper explores some of the main issues in geometric computations and the methods that have been proposed to handle roundoff errors and focuses on one method and applies it to a general iterative intersection problem.
Consistent Digital Curved Rays and Pseudoline Arrangements
TLDR
This paper extends Subject Classification Theory of computation of computation to families of curved rays going through the origin, and considers some psudoline arrangements obtained as unions of such families of rays.
Topology-Oriented Implementation—An Approach to Robust Geometric Algorithms
TLDR
This paper presents an approach, called the ``topology-oriented approach,'' to numerically robust geometric algorithms that guarantees robustness of the algorithm because combinatorial and topological computation is never contaminated with numerical errors.
On Finite-Precision Representations of Geometric Objects
Computational geometry
Consider the point p and the line l arranged in the plane as illustrated in Fig. 1.1. The following question immediately comes to mind. Does the point p lie on, above or below l? This question asks…
Applied Computational Geometry: Towards Robust Solutions of Basic Problems
A Paradigm for the Robust Design of Algorithms for Geometric Modeling
TLDR
The approach described here is a general paradigm for handling the accuracy problem for a large set of geometric algorithms, and brings together ideas and techniques from interval arithmetic, constraint management, randomization, and algebraic geometry.
Robust Geometric Computing in Motion
TLDR
A discussion of the gap between the theory and practice of geometric algorithms, together with a brief review of existing solutions to some of the problems that this dichotomy brings about, and an overview of the CGAL project and library.
The same-object problem for polyhedral solids
  • M. Karasick
  • Mathematics
    Comput. Vis. Graph. Image Process.
  • 1989
...
...

References

SHOWING 1-10 OF 27 REFERENCES
Cell Graphs: A Provable Correct Method for the Storage of Geometry *
TLDR
An approach in which topological relations are separately recorded and independent of metric positions, and it is shown that in such a geometric configuration topology is invariant to affine transformations, independently of the method selected for recording metric information.
A note on discrete representation of lines
TLDR
Equivalence classes of identically drawn lines are described in terms of Farey series; this treatment notably simplifies previous work of Dorst and Smeulders.
Algorithms for Reporting and Counting Geometric Intersections
TLDR
Algorithms that count the number of pairwise intersections among a set of N objects in the plane and algorithms that report all such intersections are given.
An Overview of Quadtrees, Octrees, and Related Hierarchical Data Structures
TLDR
An overview of hierarchical data structures for representing images, such as the quadtree and octree, is presented, based on the principle of recursive decomposition, with a greater emphasis on region data and to a lesser extent on point, line, and 3-dimensional data.
Graphs, surfaces, and homology
Preface to the third edition Preface to the first edition List of notation Introduction 1. Graphs 2. Closed surfaces 3. Simplicial complexes 4. Homology groups 5. The question of invariance 6. Some…
Finite-Resolution Computational Geometry.
  • IEEE Symp. on Foundations of Computer Science,
  • 1986
International symposium on spatial data handling
  • 1984
TABLE OF CONTENTS 1. Introduction
  • TABLE OF CONTENTS 1. Introduction
Problems using envelopes
  • Problems using envelopes
Revised Approach: Zero Order Intersection
  • Revised Approach: Zero Order Intersection
...
...