The geometry of trifocal curves with applications in architecture, urban and spatial planning

  • Maja Petrović University of Belgrade, Faculty of Transport and Traffic Engineering, Belgrade, Serbia
  • Bojan Banjac University of Belgrade, Faculty of Electrical Engineering, Belgrade, Serbia and University of Novi Sad, Faculty of technical sciences – Computer Graphics Chair, Novi Sad, Serbia
  • Branko Malešević University of Belgrade, Faculty of Electrical Engineering, Belgrade, Serbia
Keywords: Fermat-Torricelli-Weber point, trifocal curve, Java applet


In this paper we consider historical genesis of trifocal curve as an optimal curve for solving the Fermat’s problem (minimizing the sum of distance of one point to three given points in the plane). Trifocal curves are basic plane geometric forms which appear in location problems. We also analyze algebraic equation of these curves and some of their applications in architecture, urbanism and spatial planning. The area and perimeter of trifocal curves are calculated using a Java application. The Java applet is developed for determining numerical value for the Fermat-Torricelli-Weber point and optimal curve with three foci, when starting points are given on an urban map. We also present an application of trifocal curves through the analysis of one specific solution in South Stream gas pipeline project.


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