# Catch Me If You Can Part 2: When Harry Met Lloyd

When a cat wants to catch a mouse, he has to solve an interesting mathematical problem – he has to construct **a curve of pursuit** (which was by the way mentioned by Leonardo long ago).

Let’s say a point M (M for mouse, of course!) performs uniform motion along a curved path. Another point C (you understand why it is named so? no? C means cat!) moves with a constant speed V (we, cats don’t like to put too much effort!) in direction М.

To formulate the equation of the line we choose the coordinate system so that M is put in ~~the center of the Universe~~ origin and C has initial coordinates С(а, 0).

Hence :

С(t+dt) = C(t) + (C(t) – M(t))/|C(t) – M(t)| * V * dt.

If

C(t) = [x(t) y(t)],

M(t) = [p(t) q(t)],

then we obtain simultaneous equations

dx/dt = V * (x-p) / sqrt( (x-p)^2 + (y-q)^2 ),

dy/dt = V * (y-q) / sqrt( (x-p)^2 + (y-q)^2 ),

or

dy/dx = (y-q) / (x-p)

with initial conditions М(t=0) = [0 0], C(t=0) = [1 0].

Let’s take the most simple case (you can’t expect too much from the mouse) – М moves parallel to the X axis with constant speed. Hence :

p(t) = 0,

q(t) = U * t,

dy/dx = (y – U*t)/x,

or

x * dy/dx = y – U*t,

At the same time the cat runs along an arc a distance s = V*t, i.e.

x * dy/dx = y – U/V * s,

After performing differentiation we get:

dy/dx + x * d2y/dx2 – dy/dx = – U/V * ds/dx.

The length of the arc we can define as :

ds = sqrt( 1+(dy/dx)^2 ) *dx,

and finally

x * d2y/dx2 = – U/V * sqrt( 1+(dy/dx)^2 ).

It’s easy, right?

We substitute u = dy/dx and separate variables:

du / sqrt( 1+u^2 ) = – U/V * dx/x,

As the result, we now have :

log( u + sqrt(1+u^2) ) = – U/V * log(x) + const.

Using initial conditions we figure out that :

const = U/V * log(a).

We can set a = 1. Having U/V = k we get the following:

y(x) = 1/2 * { (1- x^(1-k) ) / (1-k) – (1- x^(1+k) ) / (1+k) }.

The first video below shows the trajectories of the Cat (green line) and the Mouse (blue line) with the yellow circles that show the function above.

So that is apparent now that the Cat knows math much better than the Mouse.

In real life mice never move along the straight lines, their trajectories are rather similar to random walking. The video 2 demonstrates this case: