Exact geometric computation in LEDA

We propose an e cient method that determines the sign of a multivariate polynomial expression with integer coe cients. This is a central operation on which the robustness of many geometric algorithms depends. The method relies on modular computations, for which comparisons are usually thought to require multiprecision. Our novel technique of recursive relaxation of the moduli enables us to carry out sign determination and comparisons by using only oating point computations in single precision. The method is highly parallelizable and is the fastest of all known multiprecision methods from a complexity p o i n t of view. We s h o w how to compute a few geometric predicates that reduce to matrix determinants. We discuss implementation e ciency, which can be enhanced by good arithmetic lters. We substantiate these claims by experimental results and comparisons to other existing approaches. This method can be used to generate robust and ecient implementations of geometric algorithms, including solid modeling, manufacturing and tolerancing, and numerical computer algebra (algebraic representation of curves and points, symbolic perturbation, Sturm sequences and multivariate resultants).

show abstract

“…Our techniques are comparable in speed or even faster than other techniques (e.g. 2,6,7]), and can easily handle arbitrarily large inputs.…”

Section: Resultsmentioning

confidence: 93%

Computing exact geometric predicates using modular arithmetic with single precision

Brönnimann

Emiris

Pan

et al. 1997

Proceedings of the Thirteenth Annual Symposium on Computational Geometry - SCG '97

View full text Add to dashboard Cite

show abstract

“…This explanation is speculative and machine-dependent, but the TWO-SUM algorithm below, which avoids a comparison at the cost of three additional floating-point operations, is usually empirically faster. 4 Of course, FAST-TWO-SUM remains faster if the relative sizes of the operands are known a priori, and the comparison can be avoided. Theorem 7 [17].…”

Section: Simple Additionmentioning

confidence: 99%

“…Nonetheless, the problem of constructed irrational values has been partly attacked by the implementation of "real" numbers in the LEDA library of algorithms [4]. Values derived from square roots (and other arithmetic operations) are stored in symbolic form when necessary.…”

Section: Related Work In Robust Computational Geometrymentioning

confidence: 99%

“…Two possibilities are the round-to-even rule, which specifies that the value should be rounded to the nearest p-bit value with an even significand, and the round-toward-zero rule. In four-bit arithmetic, 10011 is rounded to 1.010 × 2 4 under the round-to-even rule, and to 1.001 × 2 4 under the round-toward-zero rule. The IEEE 754 standard specifies round-to-even tiebreaking as a default.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Adaptive Precision Floating-Point Arithmetic and Fast Robust Geometric Predicates

Shewchuk¹

1997

Discrete Comput Geom

351

View full text Add to dashboard Cite

Abstract. Exact computer arithmetic has a variety of uses, including the robust implementation of geometric algorithms. This article has three purposes. The first is to offer fast software-level algorithms for exact addition and multiplication of arbitrary precision floating-point values. The second is to propose a technique for adaptive precision arithmetic that can often speed these algorithms when they are used to perform multiprecision calculations that do not always require exact arithmetic, but must satisfy some error bound. The third is to use these techniques to develop implementations of several common geometric calculations whose required degree of accuracy depends on their inputs. These robust geometric predicates are adaptive; their running time depends on the degree of uncertainty of the result, and is usually small. These algorithms work on computers whose floating-point arithmetic uses radix two and exact rounding, including machines complying with the IEEE 754 standard. The inputs to the predicates may be arbitrary single or double precision floating-point numbers. C code is publicly available for the two-dimensional and three-dimensional orientation and incircle tests, and robust Delaunay triangulation using these tests. Timings of the implementations demonstrate their effectiveness.

show abstract

“…See the recent survey papers by Yap and Dubé [49], [50]. Exact arithmetic is offered by the geometric software packages LEDA [13], [33] and CGAL [23], [38].…”

Section: Introductionmentioning

confidence: 99%

FIST: Fast Industrial-Strength Triangulation of Polygons

Held¹

2001

Algorithmica

View full text Add to dashboard Cite

We discuss a triangulation algorithm that is based on repeatedly clipping ears of a polygon. The main focus of our work was on designing and engineering an algorithm that is (1) completely reliable, (2) easy to implement, and (3) fast in practice. The algorithm was implemented in ANSI C, based on floating-point arithmetic. Due to a series of heuristics that get applied as a back-up for the standard ear-clipping process if the code detects deficiencies in the input polygon, our triangulation code can handle any type of polygonal input data, be it simple or not. Based on our implementation we report on different strategies (geometric hashing, boundingvolume trees) for speeding up the ear-clipping process in practice. The code has been tuned accordingly, and cpu-time statistics document that it tends to be faster than other popular triangulation codes. All engineering details that ensure the reliability and efficiency of the triangulation code are described in full detail. We also report experimental data on how different strategies for avoiding sliver triangles affect the cpu-time consumption of the algorithm. Our code, named FIST as an acronym for fast industrial-strength triangulation, forms the core of a package for triangulating the faces of three-dimensional polyhedra, and it has been successfully incorporated into several industrial graphics packages, including an implementation for Java 3D by Sun Microsystems.

show abstract

Exact geometric computation in LEDA

Cited by 35 publications

References 4 publications

Computing exact geometric predicates using modular arithmetic with single precision

Computing exact geometric predicates using modular arithmetic with single precision

Adaptive Precision Floating-Point Arithmetic and Fast Robust Geometric Predicates

FIST: Fast Industrial-Strength Triangulation of Polygons

Contact Info

Product

Resources

About