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 ('ray' or 'spoke') and the Greek dromos ('running' or 'racetrack'), 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 Vt. The position of the hare at time zero is (Ax, Ay) with Ax > 0 and at time t it is
The dog runs with the constant speed Vd 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 (2) and (3), it follows that
Multiplying both sides with T x − x {\displaystyle T_{x}-x} and taking the derivative with respect to x, using that
one gets
or
From this relation, it follows that
where B is the constant of integration determined by the initial value of y' at time zero, y' (0)= sinh(B − (Vt /Vd) lnAx), i.e.,
From (8) and (9), it follows after some computation that
Furthermore, since y(0)=0, it follows from (1) and (4) that
If, now, Vt ≠ Vd, relation (10) integrates to
where C is the constant of integration. Since again y(0)=0, it's
The equations (11), (12) and (13), then, together imply
If Vt = Vd, relation (10) gives, instead,
Using y(0)=0 once again, it follows that
The equations (11), (15) and (16), then, together imply that
If Vt < Vd, it follows from (14) that
If Vt ≥ Vd, one has from (14) and (17) that lim x → A x y ( x ) = ∞ {\displaystyle \lim _{x\to A_{x}}y(x)=\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 Nahin, Paul J. (2012), Chases and Escapes: The Mathematics of Pursuit and Evasion, Princeton: Princeton University Press, ISBN 978-0-691-12514-5. Gomes Teixera, Francisco (1909), Imprensa da universidade (ed.), Traité des Courbes Spéciales Remarquables, vol. 2, Coimbra, p. 255{{citation}}: CS1 maint: location missing publisher (link)
