Algebraic methods and arithmetic filtering for exact predicates on circle arcs

Devillers, Olivier; Fronville, Alexandra; Mourrain, Bernard; Teillaud, Monique

doi:10.1016/s0925-7721(01)00050-5

Cited by 26 publications

(29 citation statements)

References 9 publications

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“…(1) and (4). Furthermore, such a solution corresponds to a unique line transversal unless αa + α ′ a ′ = βb + β ′ b ′ and that point is either coplanar with lines C and D or on C or D.…”

Section: Notation and Preliminariesmentioning

confidence: 99%

“…In fact it is more efficient to first attempt to locate the roots of Equation (6), and then only resolve Lemma 9 (i.e., evaluate the discriminant ∆) when the root location is not complete [4,7]. Precisely, as we show below, we need to evaluate ∆ only to distinguish between some cases where Equation (6) has no real roots and the case where it has two positive real roots, other cases e.g.…”

Section: Lemma 10 Given a Line Segment And Three Lines There Is A Prmentioning

confidence: 99%

“…3 are the edges of the triangle) to locate the stabbing lines with respect to the triangle, but we cannot eventually compute the coordinates of this point which involve square roots. Equivalently, we can compute M k the intersection of the plane defined by Equations (1), (4) and (17 k ) and M + ∞ the direction of the line defined by Equations (1) and (4). Notice that all points S 1 , S 2 , M 1 , M 2 and M 3 are on the same line in αβζ space whose direction is M + ∞ .…”

Section: Proofmentioning

confidence: 99%

See 2 more Smart Citations

Predicates for line transversals to lines and line segments in three-dimensional space

Devillers

Glisse²,

Lazard

2008

Proceedings of the Twenty-Fourth Annual Symposium on Computational Geometry

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International audienceWhen an observer is in a 3D scene, a topological change in the view arises when the line of sight is tangent to four objects. If we consider polyhedral scenes, the relevant lines of sight are ransversals to some edges of the polyhedra. In this paper we investigate predicates about visibility events arising in this context. Namely, we consider the predicates for counting the number of line transversals to lines and segments in 3D and the predicate for determining whether a line of sight is intersected by a triangle. We also consider a predicate that order these visibility events in the rotating plane-sweep algorithm of Brönnimann et al. (2007) We present a new approach for solving these predicates and show that the degree of the resulting procedures are significantly smaller than the naive approach based on Plücker coordinates. All the degrees are considered here in the Cartesian coordinates of the points defining the lines and segments. Precisely, we present a procedure of degree 12 (resp. 15) for determining the number of transversals to four (resp. five or more) segments. We present procedures of degree 15 for the occlusion predicate and of degree 36 for the ordering predicate. In comparison, the degree of the standard procedure based on the Plücker coordinates for solving these predicates range from 36 to 168 [Everett et al. 2006]

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Section: Notation and Preliminariesmentioning

confidence: 99%

Section: Lemma 10 Given a Line Segment And Three Lines There Is A Prmentioning

confidence: 99%

Section: Proofmentioning

confidence: 99%

See 1 more Smart Citation

Predicates for line transversals to lines and line segments in three-dimensional space

Devillers

Glisse²,

Lazard

2008

Proceedings of the Twenty-Fourth Annual Symposium on Computational Geometry

Self Cite

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“…8 So for instance, the sign sequence +0 − 0 has no sign permanence, nor sign change (because there are no consecutive non-zero terms). For the sign sequence +00−, the Sturm query is computed as for ++: one permanence, no change, this gives 1.…”

Section: Such Objects Are Called Combinantsmentioning

confidence: 99%

Equations, inequations and inequalities characterizing the configurations of two real projective conics

Briand

2007

AAECC

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Abstract. Couples of proper, non-empty real projective conics can be classified modulo rigid isotopy and ambient isotopy. We characterize the classes by equations, inequations and inequalities in the coefficients of the quadratic forms defining the conics. The results are well-adapted to the study of the relative position of two conics defined by equations depending on parameters.MSC2000: 13A50 (invariant theory), 13J30 (real algebra).

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“…Predicates for arrangements of circular arcs that reduce all computations to sign determination of polynomial expressions in the input data are treated by Devillers et al [13]. Recent work by Emiris and Tsigardias [16] discusses some predicates on conics in this style; see also [15] in this volume.…”

Section: Previous Workmentioning

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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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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Algebraic methods and arithmetic filtering for exact predicates on circle arcs

Cited by 26 publications

References 9 publications

Predicates for line transversals to lines and line segments in three-dimensional space

Predicates for line transversals to lines and line segments in three-dimensional space

Equations, inequations and inequalities characterizing the configurations of two real projective conics

Complete, exact, and efficient computations with cubic curves

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