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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

See also similar curves in 3D.
 

Did the anonymous Ivorian artist who designed this engraving think they were tracing pursuit curves?


 
 
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© Robert FERRÉOL, Alain ESCULIER  2021