A Strong and Easily Computable Separation Bound for Arithmetic Expressions Involving Radicals

The Bentley-Ottmann sweep-line method can be used to compute the arrangement of planar curves provided a number of geometric primitives operating on the curves are available. We discuss the mathematics of the primitives for planar algebraic curves of degree three or less and derive efficient realizations. As a result, we obtain a complete, exact, and efficient algorithm for computing arrangements of cubic curves. Conics and cubic splines are special cases of cubic curves.The algorithm is complete in that it handles all possible degeneracies including singularities. It is exact in that it provides the mathematically correct result. It is efficient in that it can handle hundreds of curves with a quarter million of segments in the final arrangement.

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“…Reasonably efficient methods [11,6] are available to compute in FRE and to compare expressions in FRE.…”

Section: Our Resultsmentioning

confidence: 99%

Complete, exact, and efficient computations with cubic curves

Eigenwillig

Kettner

Schömer

et al. 2004

Proceedings of the Twentieth Annual Symposium on Computational Geometry

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“…Line 3 gives the time to evaluate the 10 random examples of Line 2 for 100 times each. [2] drops to 8L + 30. On the other hand, when x, y are L-bit integers, the Bit-Bound function for all three methods is the same and equal to 7.5L+30.…”

Section: Introductionmentioning

confidence: 94%

“…The expression (10) is non-unique. Indeed, there is some leeway for designing a suitable set of ul [2]-parameters for α because in general the best choice is not easily given by a fixed rule. Thus, if (u , , v) is a set of ul[2]-parameters for α, then so is either (u 2 v , , 0) or (u , 2 −v , 0), depending on whether v ≥ 0 or not.…”

Section: Generalization Of Bfmssmentioning

confidence: 99%

Constructive root bound for <tt>k</tt>-ary rational input numbers

Pion¹,

Yap²

2003

Proceedings of the Nineteenth Conference on Computational Geometry - SCG '03

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Guaranteeing accuracy is the critical capability in Exact Geometric Computation, an important paradigm for constructing robust geometric algorithms. Constructive root bounds is the fundamental technique needed to achieve such guaranteed accuracy. Current bounds are overly pessimistic in the presence of general rational input numbers. In this paper, we introduce a method which greatly improves the known bounds for k-ary rational input numbers. Since a majority of input numbers in scientific and engineering applications are either binary (k = 2) or decimal (k = 10), our results could lead to a significant speedup for a large class of applications. We apply our method to two of the best available constructive root bounds, the BFMSS Bound and the Degree-Measure Bound. Implementation and experimental results based on the Core Library are reported.

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“…For graphs of bounded degree, the polynomials have bounded degree, and the condition can be tested in polynomial time in the classical Turing machine model, with rational edge lengths as inputs, using separation bounds for algebraic computations; see [3,4,21]. We may also allow square roots of rationals as inputs.…”

Section: Triangulated Graphsmentioning

confidence: 99%

Planar Embeddings of Graphs with Specified Edge Lengths

Cabello¹,

Demaine²,

Rote³

2007

JGAA

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We consider the problem of finding a planar straight-line embedding of a graph with a prescribed Euclidean length on every edge. There has been substantial previous work on the problem without the planarity restrictions, which has close connections to rigidity theory, and where it is easy to see that the problem is NP-hard. In contrast, we show that the problem is tractable-indeed, solvable in linear time on a real RAM-for straight-line embeddings of planar 3-connected triangulations, even if the outer face is not a triangle. This result is essentially tight: the problem becomes NP-hard if we consider instead straight-line embeddings of planar 3-connected infinitesimally rigid graphs with unit edge lengths, a natural relaxation of triangulations in this context.

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A Strong and Easily Computable Separation Bound for Arithmetic Expressions Involving Radicals

Cited by 37 publications

References 11 publications

Complete, exact, and efficient computations with cubic curves

Complete, exact, and efficient computations with cubic curves

Constructive root bound for <tt>k</tt>-ary rational input numbers

Planar Embeddings of Graphs with Specified Edge Lengths

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