Ring of mixed characteristic

In geometry, a radiodrome is the path followed by a point which is pursuing another 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.

It is also the name of a weekly radio show hosted by internet personality Brad Jones and Josh Hadley.

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:

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 have to change his heading continuously. The speed of the dog is 20% faster than the speed of the hare.

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 $V_t$. The position of the hare at time zero is $(A_x,A_y)$ and at time $t$ it is

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The dog runs with the constant speed $V_d$ towards the momentary position of the hare. The differential equation corresponding to the movement of the dog, $(x(t),y(t))$, is consequently

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It is possible to obtain a closed form analytical expression $y=f(x)$ for the motion of the dog

From (Template:EquationNote) and (Template:EquationNote) follows that

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Multiplying both sides with $T_x-x$ and taking the derivative with respect to $x$ using that

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one gets

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or

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From this relation follows that

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where $B$ is the constant of integration that is determined by the initial value of $y^{\prime }$ at time zero, i.e.

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From (Template:EquationNote) and (Template:EquationNote) follows after some computations that

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If now $V_t\ne V_d$ this relation is integrated as

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where $C$ is the constant of integration.

If $V_t=V_d$ one gets instead

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If $V_t<V_d$ one gets from (Template:EquationNote) that

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In the case illustrated in the figure above $\frac{V_t}{V_d}=\frac{1}{1.2}$ and the chase starts with the hare at position $(A_x,-0.6A_x)$ what means that $y^{\prime }(0)=-0.6$. From (Template:EquationNote) one therefore gets hat the hare is caught at position $(A_x,1.21688A_x)$ and consequently that the hare will run the total distance $(1.21688+0.6)A_x$ before being caught.

If $V_t\ge V_d$ one gets from (Template:EquationNote) and (Template:EquationNote) that $\underset{x\to A_x}{lim}y(x)=\mathrm{\infty }$ what means that the hare never will be caught whenever the chase starts.