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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.unibayreuth.de/material/verfolgung/node5.html Animation by G. Tulloue: www.sciences.univnantes.fr/sites/genevieve_tulloue/Meca/Cinematique/4mouches.php 
When n points M_{1},M_{2},…,M_{n}
(traditionally, flies, mice, ladybirds...) chase one another at the same
constant speed, with M_{k} chasing
M_{k+1}
(and M_{n} chasing M_{1}),
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.






Demonstration in the case of a square.
The flies constantly form an EFGH square with the
coordinates shown in the figure.

Figure : Elisabeth Busser 


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