Apollonius’ Problem: A Study of Solutions and Their Connections

Gisch, David; Ribando, Jason M.

doi:10.33697/ajur.2004.010

Cited by 8 publications

(5 citation statements)

References 9 publications

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“…The target XY position can be obtained from the three distance ratios utilizing the Apollonius circle concept [10]. An Apollonius circle is a family of points which exhibit the following property: the ratio of the distances from any point on the Apollonius circle to two reference points is the same (shown in the Fig.…”

Section: Methodology Outlinementioning

confidence: 99%

Target signature agnostic tracking with an ad-hoc network of omni-directional sensors

2011

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Ad-hoc networks of simple, omni-directional sensors present an attractive solution to low-cost, easily deployable, fault tolerant target tracking systems. In this paper, we present a tracking algorithm that relies on a real time observation of the target power, received by multiple sensors. We remove target position dependency on the emitted target power by taking ratios of the power observed by different sensors, and apply the natural logarithm to effectively transform to another coordinate system. Further, we derive noise statistics in the transformed space and demonstrate that the observation in the new coordinates is linear in the presence of additive Gaussian noise. We also show how a typical dynamic model in Cartesian coordinates can be adapted to the new coordinate system. As a consequence, the problem of tracking target position with omni-directional sensors can be adapted to the conventional Kalman filter framework. We validate the proposed methodology through simulations under different noise, target movement, and sensor density conditions.

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Section: Methodology Outlinementioning

confidence: 99%

Target signature agnostic tracking with an ad-hoc network of omni-directional sensors

2011

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“…Void Distribution by Tangent Sphere Construction. We consider a 3D generalization for the problem of Apollonius that originally concerned the tangency of objects in the plane. Given four spherical balls ℬ i ∈ℝ 3 with radius R i and centered at r i = ( x i , y i , z i ) for i = 1, 2, 3, 4, we require the “osculating spherical ball” ℬ that is tangent to each ℬ i and remains external to all the ℬ i satisfying This corresponds to exactly one of the 2 4 (possibly degenerate) solutions of the generalized Apollonius problem.…”

Section: Methodsmentioning

confidence: 99%

Quantitative Imaging of Aggregated Emulsions

et al. 2006

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Noise reduction, restoration, and segmentation methods are developed for the quantitative structural analysis in three dimensions of aggregated oil-in-water emulsion systems imaged by fluorescence confocal laser scanning microscopy. Mindful of typical industrial formulations, the methods are demonstrated for concentrated (30% volume fraction) and polydisperse emulsions. Following a regularized deconvolution step using an analytic optical transfer function and appropriate binary thresholding, novel application of the Euclidean distance map provides effective discrimination of closely clustered emulsion droplets with size variation over at least 1 order of magnitude. The a priori assumption of spherical nonintersecting objects provides crucial information to combat the ill-posed inverse problem presented by locating individual particles. Position coordinates and size estimates are recovered with sufficient precision to permit quantitative study of static geometrical features. In particular, aggregate morphology is characterized by a novel void distribution measure based on the generalized Apollonius problem. This is also compared with conventional Voronoi/Delauney analysis.

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“…Step 3. The smallest enclosing ball coincides with the Apollonius ball that is internally tangent to Ω i for i ∈ I; see, e.g., [19] and the references therein.…”

Section: Casementioning

confidence: 99%

“…Step 3. The smallest enclosing ball coincides with the Apollonius ball that is internally tangent to Ω i for i ∈ I; see, e.g., [19] and the references therein. We say that two balls strictly intersect if they intersect at more than one points, and tangentially intersect if they intersect each other at exactly one point.…”

Section: Three-ball Generalized Sylvester Problem and The Problem Of ...mentioning

confidence: 99%

Constructions of Solutions to Generalized Sylvester and Fermat–Torricelli Problems for Euclidean Balls

Nam¹,

Hoàng²,

An³

2013

J Optim Theory Appl

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The classical Apollonius' problem is to construct circles that are tangent to three given circles in a plane. This problem was posed by Apollonius of Perga in his work "Tangencies". The Sylvester problem, which was introduced by the English mathematician J.J. Sylvester, asks for the smallest circle that encloses a finite collection of points in the plane. In this paper, we study the following generalized version of the Sylvester problem and its connection to the problem of Apollonius: given two finite collections of Euclidean balls in R n , find the smallest Euclidean ball that encloses all of the balls in the first collection and intersects all of the balls in the second collection. We also study a generalized version of the Fermat-Torricelli problem stated as follows: given two finite collections composed of three Euclidean balls in R n , find a point that minimizes the sum of the farthest distances to the balls in the first collection and shortest distances to the balls in the second collection.

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Apollonius’ Problem: A Study of Solutions and Their Connections

Cited by 8 publications

References 9 publications

Target signature agnostic tracking with an ad-hoc network of omni-directional sensors

Target signature agnostic tracking with an ad-hoc network of omni-directional sensors

Quantitative Imaging of Aggregated Emulsions

Constructions of Solutions to Generalized Sylvester and Fermat–Torricelli Problems for Euclidean Balls

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