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 Latin word radius (Eng. ray; spoke) and the Greek word dromos (Eng. running; racetrack), for there is a radial component in its kinematic analysis. 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 Template:Math. The position of the hare at time zero is Template:Math with Template:Math and at time Template:Mvar it is
Template:NumBlk
The dog runs with the constant speed Template:Math towards the instantaneous position of the hare.
The differential equation corresponding to the movement of the dog, Template:Math, is consequently
Template:NumBlk Template:NumBlk
It is possible to obtain a closed-form analytic expression Template:Math for the motion of the dog.
From (Template:EquationNote) and (Template:EquationNote), it follows that
Template:NumBlk
Multiplying both sides with and taking the derivative with respect to Template:Mvar, using that
Template:NumBlk
one gets
Template:NumBlk
or
Template:NumBlk
From this relation, it follows that
Template:NumBlk
where Template:Mvar is the constant of integration determined by the initial value of Template:Mvar' at time zero, Template:Math, i.e.,
Template:NumBlk
From (Template:EquationNote) and (Template:EquationNote), it follows after some computation that
Template:NumBlk
Furthermore, since Template:Math, it follows from (Template:EquationNote) and (Template:EquationNote) that
Template:NumBlk
If, now, Template:Math, relation (Template:EquationNote) integrates to
Template:NumBlk
where Template:Mvar is the constant of integration. Since again Template:Math, it's
Template:NumBlk
The equations (Template:EquationNote), (Template:EquationNote) and (Template:EquationNote), then, together imply
Template:NumBlk
If Template:Math, relation (Template:EquationNote) gives, instead,
Template:NumBlk
Using Template:Math once again, it follows that Template:NumBlk The equations (Template:EquationNote), (Template:EquationNote) and (Template:EquationNote), then, together imply that
Template:NumBlk
If Template:Math, it follows from (Template:EquationNote) that
Template:NumBlk
If Template:Math, one has from (Template:EquationNote) and (Template:EquationNote) that , which means that the hare will never be caught, whenever the chase starts.