## Surface Intersection Using Affine Arithmetic (1996)

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Venue: | In Graphics Interface |

Citations: | 18 - 7 self |

### BibTeX

@INPROCEEDINGS{Figueiredo96surfaceintersection,

author = {Luiz Henrique De Figueiredo},

title = {Surface Intersection Using Affine Arithmetic},

booktitle = {In Graphics Interface},

year = {1996},

pages = {168--175}

}

### Years of Citing Articles

### OpenURL

### Abstract

We describe a variant of a domain decomposition method proposed by Gleicher and Kass for intersecting and trimming parametric surfaces. Instead of using interval arithmetic to guide the decomposition, the variant described here uses affine arithmetic, a tool recently proposed for range analysis. Affine arithmetic is similar to standard interval arithmetic, but takes into account correlations between operands and sub-formulas, generally providing much tighter bounds for the computed quantities. As a consequence, the quadtree domain decompositions are much smaller and the intersection algorithm runs faster. keywords: surface intersection, trimming surfaces, range analysis, interval analysis, CAGD.

### Citations

157 |
Geometric and Solid Modeling: An Introduction
- Hoffmann
- 1989
(Show Context)
Citation Context ...ding the parameter domain uniformly into many small rectangles. However, intersecting such fine polygonal approximation is itself a difficult task. Even if we do not care about geometric degeneracies =-=[7, 8]-=-, this is a high complexity task: If there are n rectangles along each main direction in parameter space, then there are n 2 faces in each polyhedron. A naive algorithm that computes the intersection ... |

132 | Interval analysis for computer graphics
- Snyder
- 1992
(Show Context)
Citation Context ...functions. Several methods based on IA have recently been proposed for solving a number of fundamental problems in computer graphics, including ray tracing [10] and approximation of implicit surfaces =-=[11, 12, 13]-=-. Those methods have become quite popular, due to their ability to handle arbitrarily complex non-polynomial surfaces, and their immunity to round-off errors. Previously, methods based on Lipschitz co... |

131 |
New Computer Methods for Global Optimization
- Ratschek, Rokne
- 1988
(Show Context)
Citation Context ...ng automatic differentiation and interval arithmetic [15]. Global optimization, which includes computing Lipschitz bounds as a special case, has recently been shown to be feasible with range analysis =-=[16, 17]-=-. However, global optimization with range analysis has barely been explored in computer graphics [12]. The main weakness of IA is that it tends to be too conservative: the computed interval for a quan... |

126 |
Automatic Differentiation: Techniques and Applications
- Rall
- 1981
(Show Context)
Citation Context ...n computer graphics, specially because range analysis can be automated (typically with IA) [4]. In particular, Lipschitz bounds can be computed using automatic differentiation and interval arithmetic =-=[15]-=-. Global optimization, which includes computing Lipschitz bounds as a special case, has recently been shown to be feasible with range analysis [16, 17]. However, global optimization with range analysi... |

96 |
Interval arithmetic and recursive subdivision for implicit functions and constructive solid geometry
- Duff
- 1992
(Show Context)
Citation Context ...functions. Several methods based on IA have recently been proposed for solving a number of fundamental problems in computer graphics, including ray tracing [10] and approximation of implicit surfaces =-=[11, 12, 13]-=-. Those methods have become quite popular, due to their ability to handle arbitrarily complex non-polynomial surfaces, and their immunity to round-off errors. Previously, methods based on Lipschitz co... |

95 | Guaranteed ray intersections with implicit surfaces
- Kalra, Barr
- 1989
(Show Context)
Citation Context ...lynomial surfaces, and their immunity to round-off errors. Previously, methods based on Lipschitz conditions (global bounds on derivatives) appeared to be promising for computer graphics applications =-=[9, 14]-=-. However, computing Lipschitz bounds is 3 a non-trivial mathematical problem that did not seem to have an automatic solution. Methods using range analysis seem to be more popular now in computer grap... |

70 |
Computer Methods for the Range of Functions
- Ratschek, Rokne
- 1984
(Show Context)
Citation Context ...ffine arithmetic instead of interval arithmetic. 3 Interval arithmetic The classical technique of interval arithmetic (IA), also known as interval analysis, provides a natural tool for range analysis =-=[4]-=-. In IA, each quantity is represented by an interval of floatingpoint numbers. Those intervals are added, subtracted, multiplied, etc., in such a way that each computed interval is guaranteed to conta... |

67 | Affine arithmetic and its applications to computer graphics
- Comba, Stolfi
- 1993
(Show Context)
Citation Context ...ce the efficiency of the decomposition. In Section 4, we briefly describe affine arithmetic, a recent technique for range analysis that generally provides much tighter bounds than interval arithmetic =-=[5]-=-. In Section 5, we review the algorithm proposed by Gleicher and Kass [3] and give some evidence that their algorithm can be improved by replacing interval arithmetic with affine arithmetic. Section 6... |

47 |
Robust ray intersection with interval arithmetic
- Mitchell
- 1990
(Show Context)
Citation Context ...arger than the class of rational polynomial functions. Several methods based on IA have recently been proposed for solving a number of fundamental problems in computer graphics, including ray tracing =-=[10]-=- and approximation of implicit surfaces [11, 12, 13]. Those methods have become quite popular, due to their ability to handle arbitrarily complex non-polynomial surfaces, and their immunity to round-o... |

44 |
Surface algorithms using bounds on derivatives
- Filip, Magedson, et al.
- 1986
(Show Context)
Citation Context ... where the surface is almost flat. Such methods are generally restricted to specific types of surfaces, because the nature of the surface can be exploited to derive efficient tests for local flatness =-=[9]-=-. The decomposition method proposed by Gleicher and Kass [3] takes a global approach for subdividing the domains, using range analysis [4]. Given a rectangle in each domain, they compute an estimate f... |

35 |
A generalized interval arithmetic
- Hansen
- 1978
(Show Context)
Citation Context ...r range estimates to be computed (see the example at the end of this section). Other approaches to the dependency problem in IA include centered forms [4] and Hansen's generalized interval arithmetic =-=[18]-=-, in which quantities are represented by affine combinations of a fixed number of intervals. As we shall see below, new noise symbols are dynamically created during a long computation with AA. As one ... |

30 | Adaptive enumeration of implicit surfaces with affine arithmetic, Comput. Graphics Forum 15 (5
- Figueiredo, Stolfi
- 1996
(Show Context)
Citation Context ... patches that can be sewn together to bound complex shapes. Systems that use implicit surfaces as modeling primitives do not suffer from this drawback, but implicit surfaces are harder to approximate =-=[2]-=-. Several methods have been proposed for solving the important problem of computing the intersection of two parametric surfaces. These methods can be classified into two major classes: continuation me... |

22 |
Surface/surface intersection. Computer Aided Geometric Design 4
- Barnhill, Farin, et al.
- 1987
(Show Context)
Citation Context ...t known to be on both surfaces, these methods build an approximation for the intersection curve by marching along the curve, successively computing a new point based on the previous point (or points) =-=[6]-=-. Continuation methods must use numerical approximations not only for marching along the curve, but also for finding starting points. Since the intersection might have several connected components, a ... |

22 |
Interval Methods in Computer Graphics
- Suffern, Fackerell
- 1991
(Show Context)
Citation Context ...functions. Several methods based on IA have recently been proposed for solving a number of fundamental problems in computer graphics, including ray tracing [10] and approximation of implicit surfaces =-=[11, 12, 13]-=-. Those methods have become quite popular, due to their ability to handle arbitrarily complex non-polynomial surfaces, and their immunity to round-off errors. Previously, methods based on Lipschitz co... |

22 | Automatic Dierentiation: Techniques and Applications - Rall - 1981 |

15 | Interval arithmetic and recursive subdivision for implicit functions and constructive solid geometry - Du - 1992 |

9 |
An interval refinement technique for surface intersection
- Gleicher, Kass
- 1992
(Show Context)
Citation Context ...s. These methods can be classified into two major classes: continuation methods and decomposition methods. In this paper, we describe a variant of a decomposition method proposed by Gleicher and Kass =-=[3]-=-. Instead of using interval arithmetic to guide the recursive domain decomposition, as they did, our variant uses affine arithmetic. Affine arithmetic is similar to standard interval arithmetic, but t... |

6 |
Local robustness and its application to polyhedral intersection
- Stewart
- 1994
(Show Context)
Citation Context ...ding the parameter domain uniformly into many small rectangles. However, intersecting such fine polygonal approximation is itself a difficult task. Even if we do not care about geometric degeneracies =-=[7, 8]-=-, this is a high complexity task: If there are n rectangles along each main direction in parameter space, then there are n 2 faces in each polyhedron. A naive algorithm that computes the intersection ... |

6 |
Global optimization using interval analysis. Number 165
- Hansen
- 1992
(Show Context)
Citation Context ...ng automatic differentiation and interval arithmetic [15]. Global optimization, which includes computing Lipschitz bounds as a special case, has recently been shown to be feasible with range analysis =-=[16, 17]-=-. However, global optimization with range analysis has barely been explored in computer graphics [12]. The main weakness of IA is that it tends to be too conservative: the computed interval for a quan... |

6 | Geometric and Solid Modeling: An Introduction - mann - 1989 |

4 |
Surfaces in computer-aided geometric design: A survey with new results. Computer Aided Geometric Design
- Barnhill
- 1985
(Show Context)
Citation Context ...y are easy to approximate and render, and there is a huge literature on special classes of surfaces suitable for shape design, such as B'ezier and splines surfaces, for which special algorithms exist =-=[1]-=-. However, using parametric surfaces for modeling solids in CSG systems requires efficient and robust methods for computing surface intersection, mainly for trimming surfaces into patches that can be ... |

1 | Adaptive enumeration of implicit surfaces with a ne arithmetic - Figueiredo, Stol - 1995 |

1 | An interval re nement technique for surface intersection - Gleicher, Kass - 1992 |