Radiodrome
In geometry, a radiodrome is a specific type of pursuit curve: the path traced by a point that continuously moves toward a target traveling in a straight line at constant speed. The term comes from the Latin radius and the Greek dromos, reflecting the radial nature of the motion.
The most classic and widely recognized example is the so-called dog curve, which describes the path of a dog swimming across a river toward a hare moving along the opposite bank. Because of the current, the dog must constantly adjust its heading, resulting in a longer, curved trajectory. This case was first described by the French mathematician and hydrographer Pierre Bouguer in 1732.
Radiodromes are distinguished from other pursuit curves by the assumption that the pursuer always heads directly toward the target’s current position, while the target moves at a constant velocity along a straight path.
Mathematical analysis
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 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 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 that
The equations, and, then, together imply that
If, it follows from that
If, one has from and that, which means that the hare will never be caught, whenever the chase starts.