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 (beam) and the Greek word dromos (running). The classical (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 food it has spotted on the other side. Because the dog drifts downwards with the current, it will have to change its heading; it will also have to swim further than if it had computed 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. It is illustrated by the following figure:
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 . The position of the hare at time zero is and at time it is
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(1) |
The dog runs with the constant speed towards the momentary position of the hare. The differential equation corresponding to the movement of the dog, , is consequently
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(2) |
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(3) |
It is possible to obtain a closed form analytical expression for the motion of the dog
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(4) |
Multiplying both sides with and taking the derivative with respect to using that
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(5) |
one gets
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(6) |
or
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(7) |
From this relation follows that
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(8) |
where is the constant of integration that is determined by the initial value of at time zero, i.e.
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(9) |
From (8) and (9) follows after some computations that
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(10) |
If now this relation is integrated as
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(11) |
where is the constant of integration.
If one gets instead
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(12) |
If one gets from (11) that
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(13) |
In the case illustrated in the figure above and the chase starts with the hare at position what means that . From (13) one therefore gets that the hare is caught at position and consequently that the hare will run the total distance before being caught.
If one gets from (11) and (12) that what means that the hare never will be caught whenever the chase starts.