Because I'm a loser, I played around with Mathematica a bit, explicitly solved the differential equation, and plotted the solution. Hope it makes things clearer... or maybe not. =)
In this picture, assume the lake has radius 10, and the monster is initially located at coordinates (-10, 0). Time is denoted by t.
As you can see, the "spiral" is actually a semicircle of diameter R/4. In this part of your path, you are moving such that the angle between you and the monster remains constant, because you are moving at its maximum angular speed.
The first part, for t < 0.025 is necessary so that the minimum distance point suddenly moves to the point opposite that of the monster, and it will start moving some direction as soon as you start moving. Here, we assume it moves counterclockwise to reach the opposite point. Then at t = 0.025, we're at a point such that the angle between you and the monster is somewhat less than 180 degrees. This is actually important to avoid ambiguities in the monster's possible movement -- at 180 degrees between you and it, it could move either direction towards you. But if less than 180 degrees, it is forced to travel the same angular direction you're travelling in.
Comments (2)
DATE: 06/16/2003
01:33:06
Just wanted to point out that r(t) = R/4 * sin 4t is a simpler solution to the differential equation.
Posted by Anonymous | September 2, 2006 6:21 AM
Posted on September 2, 2006 06:21
DATE: 06/16/2003
01:34:40
Er, assuming you ignore that first little 0.025 increment, that is.
Posted by Anonymous | September 2, 2006 6:21 AM
Posted on September 2, 2006 06:21