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.
Introduce a coordinate system with origin at the position of the dog at timezero and with y-axis in the direction the hare is running with the constantspeed . 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
Tx-x
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 thatThe equations, and, then, together imply that
If, it follows from that If, one has from and that
| \lim | |
| x\toAx |
y(x)=infty