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 $V_{t}$. The position of the hare at time zero is $(A_{x}, A_{y})$ with $A_{x} > 0$ and at time $t$ it is The dog runs with the constant speed $V_{d}$ towards the instantaneous position of the hare.

The differential equation corresponding to the movement of the dog, $(x(t), y(t))$, is consequently

It is possible to obtain a closed-form analytic expression $y=f(x)$ for the motion of the dog. From ($$) and ($$), it follows that

Multiplying both sides with $$T_x-x$$ 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, $y' (0)= sinh(B − (V_{t} /V_{d}) lnA_{x})$, i.e.,

From ($$) and ($x$), it follows after some computation that

Furthermore, since $y(0)=0$, it follows from ($$) and ($$) that

If, now, $V_{t} ≠ V_{d}$, relation ($$) integrates to where $$ is the constant of integration. Since again $y(0)=0$, it's

The equations ($B$), ($y$) and ($$), then, together imply

If $V_{t} = V_{d}$, relation ($$) gives, instead, Using $y(0)=0$ once again, it follows that

The equations ($$), ($$) and ($$), then, together imply that

If $V_{t} < V_{d}$, it follows from ($$) that If  $V_{t} ≥ V_{d}$,  one has from ($$) and ($$) that $$\lim_{x \to A_x}y(x) = \infty$$,  which means that the hare will never be caught, whenever the chase starts.