Monday 15 August 2016

Using Circle Geometry to Hunt Pokemon

If you know me in real life, it will come as no surprise to you that I have spent most of my summer holiday so far playing Pokemon Go. Those of you who don't know me in real life, if you see a pink-haired twenty-something with a phone precariously attached to a Pikachu lanyard wearing dungarees (optimal number of pockets) and Go Walk Sketchers (optimal walking shoe) walking at as close to 7km per hour as possible (optimal PoGo speed) then that's me.

After a few weeks of aimlessly walking around Coventry city centre or sitting on the steps of the bank where some kind stranger will always drop a lure, I decided that I had caught enough Drowzee and would like to make a bit of progress towards catching 'em all. My pokedex was hovering around the 80 mark for a long time, despite making trips to Leeds, Manchester and Birmingham in the hopes of finding exotic local pokemon and hatching eggs like Bernard Matthews. So I decided to give pokemon tracking a try.

In the bottom right-hand corner of the screen when you're in map view, there is a window you can expand that says "sightings" and lists some pokemon that have spawned nearby. Sometimes I would see a rare pokemon on this window, but learned helplessness has taught me that the rare ones never pop up when you want them to. By "pop up" I mean appear on your map as a tappable, catchable pokemon. So I decided that instead of wandering aimlessly and hoping for the rare ones to pop up, I would hunt them down strategically using geometry.

A pokemon will appear in "sightings" when you are within 200m of it. The pokemon will be catchable when you are within 70m of it. So when a rare pokemon, let's say a Charizard, appears on your sightings, you should be picturing the following diagram:

Of course, you could be anywhere in the purple circle (but not in the pink circle, or the Charizard would already be catchable). The fact that the pokemon just appeared in your sightings could either mean that you have just stepped inside the purple circle, or the pokemon has just spawned. Pokemon disappear after 15 minutes, so what you do next needs to be efficient and at a bit of a jog if possible. If you have a buddy with you, this is much easier, as you'll see in a minute.

Let's call the point you're at point A. What you need to do is identify a straight path that you can walk along that goes in both directions from point A. This can be very difficult, depending where you are. I have found that it is much easier in a park than in the city centre. Now, walk along that path, remembering where you started. Ideally, you would count your paces as you walk. Keep walking until the Charizard disappears from your sightings. The point where that happens we'll call point B. This will be a point on the circumference of the circle in your head. Of course, you could get lucky and walk right into the pink circle, in which case, get the razz berries and ultra balls ready! But let's assume the unluckiest situation.

Next, you need to turn around and walk in the exact opposite direction, back to point A and beyond it until the Charizard disappears from your sightings again. Call this point C. If you have a buddy with you, they can do this bit whilst you are doing step one, to save time. Again, it would be good if you could count your paces.

Now that you have identified two points on the circumference of the circle, and have walked a chord of the circle, this is where the geometry comes in. The perpendicular bisector of any chord of a circle always passes through the centre of the circle.

If you don't believe me, think about this: take a random chord of a circle and join up its end points to the centre as in this diagram:

You should be able to see that this makes an isosceles triangle, because two of the sides are radii.

This means that this triangle has a line of symmetry and if you cut it down this line, you get two right-angled triangles:

And clearly this line of symmetry goes through the centre of the circle.

So, back to our hunt. We're at point C, and we need to find the perpendicular bisector of the line segment BC which is the path we have just walked. The first thing we need to do is find the midpoint of B and C. In some places it is easy to do this by eye, if you have a good map or if you're in a very flat area. But if you have counted your paces, you will be able to find the midpoint much more accurately. I personally don't bother with counting. Because we're only trying to get inside the pink circle, not get to the exact centre, we don't have to be that accurate. So, walk to this midpoint, which we'll call D.

Now, turn ninety degrees and walk. But Emma! (I hear you cry) There's two ways of turning ninety degrees! Yes, you're right. At this point, you have not uniquely defined the purple circle. If you draw two random dots on a page, there are always exactly two different circles with a given radius that pass through those two points. You don't know which circle it is, so you have to guess. So turn ninety degrees in any direction and start walking (or running!) until one of two things happen: the Charizard pops up and you catch it, or the Charizard disappears from your sightings, in which case you do a 180 degree turn and run for it! If you have a buddy with you, you can take one direction each and invent some kind of signal for "I found it!" (smoke signal? A whistle? Make a sound like a dying giraffe?)

You have to do all this in the space of 15 minutes which can be tricky, and sometimes involves running and looking like a loon. But me and my husband went out to Coombe Abbey country park and Coventry's War Memorial Park last week and managed to use this method successfully several times. The handy thing is, pokemon can't spawn just anywhere, there are a set number of spawn points in a given area. So once we identified some of the spawn points, we didn't even have to use the method, we could run to the nearby spawn point we found earlier. There are still enough different spawn points around to keep the game challenging though.

In case you were wondering, my Pokedex is now up to 94. I promised my tutor group that I would have caught 'em all before term starts again in September, which is looking very unlikely. Then again, my tutor group promised me they wouldn't fail their AS levels, so I might have a bit of leverage there...

I hope this method helps you hunt down some rare pokemon and maybe understand the relevance of circle geometry a bit more. (Oh my gosh, did I just come up with a "real life" application of geometry??? Noooooo! Keep Maths pure, people!)

Emma x x x