No more tigers and rabbits...

It's somewhat funny how I/Dr. Manne keep getting distracted. Although I have created several Mathematica "programs" (I can't decide whether or not Mathematica counts as legitimate programming) that simulate various situations related to tigers chasing rabbits, after some discussion we've decided to mostly forget about the Homicidal Chauffeur Problem. The reasons for this are as follows: 1) after some investigation, we have realized that this problem and its variations have been so well researched that it is really pointless to continue any further 2) optical trapping is cooler 3) the respective optimal (as far as we could tell) strategies for the prey and predator just are not that interesting.

Nonetheless, I will explain a little bit about this problem and about these strategies.

Problem:
There is tiger. It wants to catch and eat the rabbit. The rabbit is some distance away from the tiger and enjoys life.

When the tiger begins chasing the rabbit, what is its best pursuit strategy if it wants to catch the rabbit as quickly as possible? Similarly, what should the rabbit do to maximize its time alive?

Let's look at the predator's strategies first...
To make this easy let's pretend that the rabbit is stupid and just runs in a circle centered about the origin and that the tiger begins at the center of this circle. Also, the tiger can make infinitely sharp turns (i.e. acceleration is infinite).

Stupid Strategy:
If the tiger is faster than the rabbit, it can eventually catch the rabbit by simply going towards where the rabbit currently is (i.e. the velocity vector for the tiger always points directly at the rabbit). This is the "stupid strategy". It works, but it is inefficient. Here is a picture (red is the rabbit's path, blue is the tiger's path):


Constant Bearing Strategy:
A much better strategy for the tiger is to go towards where it predicts the rabbit will be later on. What this means is that the tiger pretends at every instant in time that the rabbit will continue going in a straight line from its current position at its current velocity.

The tiger then calculates a path (a line) that will intercept the rabbit's line and begins to move along it. This is known as the constant bearing strategy. For a rabbit running in a circle with the tiger starting at the center of the circle, this strategy will allow the tiger to catch the rabbit in about 1/3 of the time than if it used the stupid strategy.

But, in this case, the rabbit is running in a circle and so the tiger doesn't move along a line. Instead, the line becomes a little bit bent. Here is a picture (red is the rabbit's path, blue is the tiger's path):


The Rabbit's Strategy:
If the tiger is faster than the rabbit and can make infinitely sharp turns, then there is no hope for the rabbit. It will be caught and killed.

But if the tiger's acceleration is "significantly smaller" than that of the rabbit (the rabbit can make sharper turns than the tiger), then the rabbit may be able to escape.

Unfortunately, the strategy is what you would expect and is pretty simple (and mostly uninteresting): the rabbit simply makes a bunch of sharp turns. It runs along a line and waits until the tiger gets "close" to it. It then makes the sharpest turn possible back towards the tiger. It then repeats this...

So that's the basics of the problem. Things get difficult with acceleration and multiple predators and or obstacles...

Anyway, so as I mentioned, instead of pursuit problems, I will continue working on optical trapping type stuff in the future.

Also, if you are interested in pursuit problems/math/fairly simple differential equations, GET THIS BOOK! It is very easy to follow (you only need to be comfortable with separable differential equations) and quite entertaining...

I would love to explain more, but this post is already looooong...