Geometrical solution to the Fermat problem with arbitrary weights
type: Item
instance of: scholarly article
title: Geometrical solution to the Fermat problem with arbitrary weights
zbMATH Open document ID: 1036.90050
DOI: 10.1023/A:1026167011686
author: Galina Jalal
author: Jakob Krarup
published in: Annals of Operations Research
publication date: 2004-01-05T00:00:00Z
review text: Given three points in the plane, the Fermat problem is to find a fourth point such that the sum of (weighted) distances to the given points is minimized. After giving a detailed survey of the related literature, geometric and analytic solutions are presented for arbitrary combinations of positive and negative weights. In the case that all weights are positive, a geometric argument related to the triangle inequality plays a central role in the construction. If, on the other hand, at least one of the weights is negative, an earlier approach of J. Krarup (1998) is extended which is based on the definition of a set of ``companion instances'' with positive weights. The results imply alternative and relatively simple correctness proofs for the constructions of Torricelli (1645), Cavalieri (1647), Viviani (1659), Simpson (1750), and Martelli (1998).
Mathematics Subject Classification ID: 90B85
zbMATH DE Number: 2021309
zbMATH Keywords: Fermat problem
zbMATH Keywords: geometrical solution
zbMATH Keywords: arbitrary weights
reviewed by: Kathrin Klamroth
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2021309