Distance of closest approach of two arbitrary hard ellipses in two dimensions

Zheng, Xiaoyu; Palffy‐Muhoray, Peter

doi:10.1103/physreve.75.061709

Cited by 73 publications

(47 citation statements)

References 17 publications

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“…It is also possible to represent the ellipse as a 2 Â 2 symmetric matrix so that the matrix is independent of the ellipse's origin (Zheng and Palffy-Muhoray 2007). By use of this representation, it is slightly easier to transform the ellipse's shape without changing its position.…”

Section: Ellipse Representationsmentioning

confidence: 99%

“…The distance of closest approach d cr of two arbitrary hard ellipses in 2D can be determined with the method of Zheng and Palffy-Muhoray (2007).…”

Section: Zpm Expansion Factormentioning

confidence: 99%

“…The closest approach between the circle and the ellipse is then found analytically, recovering the closest approach between the ellipses E 1 and E Ã 2 by applying the inverse of the transformation used to obtain the circle (Step 3). Complete details of these calculations are explained in Zheng and Palffy-Muhoray (2007), including the solution of the resulting quartic equation. Now, we make use of the problem's symmetry to determine the expansion factor (Step 4).…”

Section: Zpm Expansion Factormentioning

confidence: 99%

See 2 more Smart Citations

Explicit Calculation of Expansion Factors for Collision Avoidance Between Two Coplanar Survey-Error Ellipses

Sawaryn

Jamieson²,

McGregor³

2013

SPE Drilling &Amp; Completion

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Summary The positional uncertainty about a point on a wellbore is commonly represented as an ellipsoid. The ellipsoid also accounts for the dimensions of the casing or open hole. With the use of this model, at any time the resulting uncertainty about a wellbore along its trajectory is a curved, continuous cone. To a good approximation, the intersection of the plane normal to a reference well with these cones can be represented as ellipses. This simple geometrical model has been adopted by standards organizations to define minimal acceptable separation distances between wellbores (e.g., the Norwegian NORSOK D-10 Standard and Oil and Gas UK Well Integrity Guidelines). Because of mathematical difficulties, the existing methods for calculating the resulting separation factors are only approximations and may be either too optimistic or too conservative, particularly for ellipses with high eccentricities. This paper presents explicit equations for determining the exact condition in which the ellipses touch, expressing the result as an expansion scale factor. Methods are presented for the expansion of either ellipse or both, together with implementation notes and other associated tools. The new algorithms are only marginally less efficient than the existing approximation methods, and they can be used to increase the allowable proximity of two adjacent wells while satisfying the geometrical and probabilistic constraints. The examples included in the paper illustrate this. The proposed calculation method is consistent with existing industry wellbore uncertainty models. Because the determination of the osculating condition is exact, the calculation is neither too optimistic nor too conservative. This paper is a response to discussions held at the SPE Wellbore Positioning Technical Section meeting on 3 November 2011.

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Section: Ellipse Representationsmentioning

confidence: 99%

“…The distance of closest approach d cr of two arbitrary hard ellipses in 2D can be determined with the method of Zheng and Palffy-Muhoray (2007).…”

Section: Zpm Expansion Factormentioning

confidence: 99%

Section: Zpm Expansion Factormentioning

confidence: 99%

See 1 more Smart Citation

Explicit Calculation of Expansion Factors for Collision Avoidance Between Two Coplanar Survey-Error Ellipses

Sawaryn

Jamieson²,

McGregor³

2013

SPE Drilling &Amp; Completion

View full text Add to dashboard Cite

show abstract

“…e �rst method in [31,32] consists of determining the intersection of the ellipsoids with the plane containing the line joining their centers and rotating the plane. e distance of closest approach [33] of the two ellipses formed by the intersection is a periodic function of the plane orientation, of which the maximum value corresponds to the closest distance between the two ellipsoids. e second method [34] is based on the discriminant of their characteristic polynomial.…”

Section: Introductionmentioning

confidence: 99%

Flocking for Multiple Ellipsoidal Agents with Limited Communication Ranges

Duc

2013

ISRN Robotics

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is paper contributes a design of distributed controllers for �ocking of mobile agents with an ellipsoidal shape and a limited communication range. A separation condition for ellipsoidal agents is �rst derived. Smooth step functions are then introduced. ese functions and the separation condition between the ellipsoidal agents are embedded in novel pairwise potential functions to design �ocking control algorithms. e proposed �ocking design results in (1) smooth controllers despite of the agents' limited communication ranges, (2) no collisions between any agents, (3) asymptotic convergence of each agent's generalized velocity to a desired velocity, and (4) boundedness of the �ock size, de�ned as the sum of all distances between the agents, by a constant.

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“…The contact term σ(r, ω 1 , ω 2 ) approximates the geometrical "contact distance" between two ellipsoids (see Refs. [97,98] for a discussion). A similar generalisation of the GB potential to nonhomogeneous biaxial interactions is that of Cleaver et al [99].…”

Section: Model Potentialsmentioning

confidence: 99%