MUTUAL PURSUIT CURVES (or fly curves)

 Problem first posed by Lucas in 1877. Paper "Polygons of pursuit", Bernhart, 1959 Wikipedia: fr.wikipedia.org/wiki/Probl%C3%A8me_des_souris Very thorough website in German: did.mat.uni-bayreuth.de/material/verfolgung/node5.html Animation by G. Tulloue: www.sciences.univ-nantes.fr/sites/genevieve_tulloue/Meca/Cinematique/4mouches.php

When n points M1,M2,,Mn (traditionally, flies, mice, ladybirds...) chase one another at the same constant speed, with Mk chasing Mk+1 (and Mn chasing M1), the trajectories of these points are mutual pursuit curves.

 To get the figure opposite, we had the differential system stemming from the n relations solved by Maple. In the case of a triangle, it can be noted that the two flies that were the furthest apart are the ones who meet first! When the initial figure is a regular polygon with the points in their apparition order, then the trajectories are logarithmic spirals with asymptotic points the centre of the polygon.
The parameter of the spiral is , and the polar tangential angle is where n is the number of sides of the polygon.
The length of the trajectory of any fly is therefore equal to where R is the radius, and the length of the side.   Case of a pentagon Case of a hexagon Case of an octagon

 Demonstration in the case of a square. The flies constantly form an EFGH square with the coordinates shown in the figure. The collinearity of the speed vector at E with , results in , hence the differential equation of the curve: . By passing in polar coordinates the equation transforms into , hence the solution , with a = 1 into applying the initial condition. A parameterization is therefore ; with this, the speed of the flies is , it decreases towards 0, and the time course is infinite, for a length (side of the starting square   ).   With , the flies have a speed   constant equal to , for a travel time equal to 1. Figure : Elisabeth Busser Case of a square, copied 4 times by symmetries Case of an equilateral triangle, copied 7 times by symmetries   Did the anonymous Ivorian artist who designed this engraving think they were tracing pursuit curves?