Radiodrome

The path of a dog chasing a hare running along a vertical straight line at a constant speed. The dog runs towards the momentary position of the hare, and will be changing his heading continuously.

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 Template:Lang (Template:Gloss or Template:Gloss) and the Greek Template:Transl (Template:Gloss or Template:Gloss), 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 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 $T_x-x$ 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 $\underset{x\to A_x}{lim}y(x)=\mathrm{\infty }$, which means that the hare will never be caught, whenever the chase starts.

See also

• Mice problem
• Tschirnhausen cubic, a special case of the radiodrome in which the pursuer moves twice as fast as its target

References

• Template:Citation.
• Template:Citation