In geometry, a radiodrome is the pursuit curve followed by a point that is pursuing another linearly-moving point. The term is derived from the Greek words and . The classic (and best-known) form of a radiodrome is known as the "dog curve"; this is the path a dog follows when it swims across a stream with a current after something it has spotted on the other side. Because the dog drifts with the current, it will have to change its heading; it will also have to swim further than if it had taken the optimal heading. This case was described by Pierre Bouguer in 1732.
A radiodrome may alternatively be described as the path a dog follows when chasing a hare, assuming that the hare runs in a straight line at a constant velocity.
Mathematical analysis
Introduce a coordinate system with origin at the position of the dog at time
zero and with y-axis in the direction the hare is running with the constant
speed . The position of the hare at time zero is with and at time it is
The dog runs with the constant speed towards the instantaneous position of the hare.
The differential equation corresponding to the movement of the dog, , is consequently
It is possible to obtain a closed-form analytic expression for the motion of the dog. From () and (), it follows that
Multiplying both sides with and taking the derivative with respect to , using that
one gets
or
From this relation, it follows that
where is the constant of integration determined by the initial value of ' at time zero, , i.e.,
From () and (), it follows after some computation that
Furthermore, since , it follows from () and () that
If, now, , relation () integrates to
where is the constant of integration. Since again , it's
The equations (), () and (), then, together imply
If , relation () gives, instead,
Using once again, it follows that
The equations (), () and (), then, together imply that
If , it follows from () that
If , one has from () and () that , which means that the hare will never be caught, whenever the chase starts.
See also
Mice problem
References
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