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 Greek words ῥᾴδιος, rhā́idios, 'easier' and δρόμος, drómos, 'running'. 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.
![Graph of a radiodrome, also known as a dog curve](../I/Dog_curve.svg.png.webp)
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
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(1)
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
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(2)
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(3)
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
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.
(4)
Multiplying both sides with and taking the derivative with respect to x, using that
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(5)
one gets
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(6)
or
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(7)
From this relation, it follows that
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(8)
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.,
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(9)
From (8) and (9), it follows after some computation that
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.
(10)
Furthermore, since y(0)=0, it follows from (1) and (4) that
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.
(11)
If, now, Vt ≠ Vd, relation (10) integrates to
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(12)
where C is the constant of integration. Since again y(0)=0, it's
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.
(13)
The equations (11), (12) and (13), then, together imply
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.
(14)
If Vt = Vd, relation (10) gives, instead,
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.
(15)
Using y(0)=0 once again, it follows that
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(16)
The equations (11), (15) and (16), then, together imply that
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.
(17)
If Vt < Vd, it follows from (14) that
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(18)
If Vt ≥ Vd, one has from (14) and (17) that , which means that the hare will never be caught, whenever the chase starts.
See also
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
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