The Generalized Fermat's Point

Tong, Jingcheng; Chua, Y. S.

doi:10.1080/0025570x.1995.11996317

Cited by 7 publications

(3 citation statements)

References 2 publications

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“…B l b , (0, ) A a . ( , ) P x y is the quasi-Fermat's Point which can make the total length of pine line shorter [5]. It is assumed that the unit costs of shared pipe or non-shared pipeline are the same.…”

Section: Oil Pipe Arrangement Scheme With Compensation To Private Pro...mentioning

confidence: 99%

Optimization of Oil Pipe Arrangement with Compensation Conditions

Wei

Zhao

Lei

2014

AMM

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Oil pipelines usually pass through suburbs areas and urban areas. When the pipeline occupy some private properties, the compensation to private properties owners is necessary. In fact, the oil pipe arrangement is very complicated when the unit cost in suburbs and urban areas is different. Hence the oil pipe arrangement scheme with compensation conditions is proposed to study the complicated problem. The model shows that compensation cost can affect the pipeline arrangement very sensitively. The model can obtain the optimal scheme to construct oil pipe when given different compensation cost.

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Section: Oil Pipe Arrangement Scheme With Compensation To Private Pro...mentioning

confidence: 99%

Optimization of Oil Pipe Arrangement with Compensation Conditions

Wei

Zhao

Lei

2014

AMM

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“…The very recent paper by Abi-Khuzam [1] gives a treatment of the weighted FermatTorricelli problem when the weights satisfy the triangle inequalities (i.e., when they can serve as the side lengths of a triangle), and it gives interesting applications to the problem and updates the references in [17]. Other treatments, besides those cited in these two publications, appear in [6,10,11,18,24], and [8,9,12,20,23,27], and [26].…”

mentioning

confidence: 99%

“…In fact, some papers even skip some of these last steps. For example, the short and nice paper of Tong [24] gives the strong, but false, impression that the point of concurrence of the lines AA , BB , and CC lies always inside ABC.…”

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confidence: 99%

A complete analytical treatment of the weighted Fermat–Torricelli point for a triangle

Hajja

Zachos

2016

J. Geom.

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The weighted Fermat-Torricelli problem with positive weights α, β, and γ asks for the point in the plane of a given triangle ABC that minimizes the function f (P ) = α P A + β P B + γ P C . This paper provides a complete, fully analytical, and self-contained solution to this problem. The solution starts, as is most natural, with the gradient equation ∇f = 0, and obtains all the desired results by some delicate algebraic manipulations of this equation.The Fermat-Torricelli problem asks about the point(s) P in the Euclidean plane R 2 whose distances from the vertices of a given triangle A 1 A 2 A 3 have a minimal sum. In other words, it asks about the point(s) P at which the functionattains its minimum, where X−Y stands for the Euclidean distance between X and Y . The weighted Fermat-Torricelli problem with positive weights w 1 , w 2 , and w 3 asks about the point(s) P at which the function(1) attains its minimum. When the weights w 1 , w 2 , and w 3 are equal, the problem reduces to the original problem, which will be sometimes referred to as the unweighted Fermat-Torricelli problem. Both the weighted and unweighted Fermat-Torricelli problems, together with variants and generalizations, have attracted a great amount of attention and have generated a wealth of research. Different treatments of both versions abound in the existing literature. The

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Uniqueness and nonuniqueness for the L1 minimization source localization problem with three measurements

Kwon

2022

Applied Mathematics and Computation

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The Generalized Fermat's Point

Cited by 7 publications

References 2 publications

Optimization of Oil Pipe Arrangement with Compensation Conditions

Optimization of Oil Pipe Arrangement with Compensation Conditions

A complete analytical treatment of the weighted Fermat–Torricelli point for a triangle

Uniqueness and nonuniqueness for the L1 minimization source localization problem with three measurements

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