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 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 formula_1. The position of the hare at time zero is with and at time it is
The dog runs with the constant speed formula_2 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 formula_3 and taking the derivative with respect to using that
or
From this relation it follows that
where is the constant of integration determined by the initial value of ' at time zero, , i.e.,
where is the constant of integration. Since again , it's
\frac{y'(0)-\sqrt{1 + \frac{V_t}{V_d}} 
The equations (), () and () then together imply
If , relation () gives instead
If , one has from () and () that formula_4, which means that the hare will never be caught, whenever the chase starts.