Radiodrome


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 lit=easier and lit=running. The classic 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 computations 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
The equations, and then together imply

If , it follows from that

If , one has from and that, which means that the hare will never be caught, whenever the chase starts.