Weberian Focal-Directorial Curves
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Weberian Focal‑Directorial Curves Geometric Genesis and Form Variation Maja Petrović1 · Branko Malešević2 · Radovan Štulić3 Accepted: 5 September 2020 © Kim Williams Books, Turin 2020
Abstract In this paper, new plane curves (multidirectorial curves as counterpart to multifocal curves; and focal-directorial curves as curves of a transitory type) are considered through geometric genesis, and their form variation for special choices of foci and directrices disposition. Furthermore, the form variation of Weberian focaldirectorial curves (WFDC) for different values of accompanying parameter S and Weberian coefficients is considered. Keywords Fermat–Torricelli point · Maxwell’s curves · Weber location problem · Geometric inequality · Weberian focal-directorial curves
Introduction Genesis of curves, nowadays, plays a significant role in architectural design due to the fact that the contemporary computer technologies enable us to solve various mathematical problems in order to create complex geometric forms being attractive and constructively applicable at the same time. Therefore, the defining proper
* Maja Petrović [email protected] Branko Malešević [email protected] Radovan Štulić [email protected] 1
The Faculty of Transport and Traffic Engineering, University of Belgrade, Vojvode Stepe 305, Belgrade, Serbia
2
School of Electrical Engineering, University of Belgrade, Bulevar Kralja Aleksandra 73, Belgrade, Serbia
3
Faculty of Technical Sciences, University of Novi Sad, Trg Dositeja Obradovića 6, Novi Sad, Serbia
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mathematical rules together with the curves properties investigation becomes of a crucial importance in generating anticipated shapes. Genesis of Focal Curves I—The Fermat geometric problem. The starting point of our research was a wellknown geometric problem stated by the French mathematician Pierre de Fermat in the seventeenth century. In original Latin, the problem is as follows: “Datis tribus punctis, quartum reperire, a quo si ducantur tres rectae ad data puncta, summa trium harum rectarum sit minima quantitas” (de Fermat 1679: 153), with the English translation being: “For three given points, the fourth is to be found, from which if three straight lines are drawn to the given points, the sum of the three lengths is minimum” (Sec. 3; Brazil et al. 2014: 4). II—The Torricelli construction of the Fermat point. The geometric solution to this problem was given by Evangelista Torricelli Torricelli et al. (1919), Viviano (1659). This Italian physicist and mathematician constructed three equilateral triangles over the sides of the initial triangle (herewith A, B, C, Fig. 1). Next step of Torricelli’s geometric solution was to draw three circumcircles of the equilateral triangles. Then the point of intersection of these circles ( c1 , c2 , c3 ) is the solution to the Fermat
Fig. 1 Geometric drawing of the Torricelli construction of the Fermat point
Weberian Focal‑Directorial Curves
problem. This point, called the Fermat–Torricelli point (F, F
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