I have a nasty feeling this is going to turn out to be one of those problems in which some environmental consideration you aren't modelling turns out to massively affect the prior probability distribution and inform the strategy.
In particular: along what shape of path were you (hypothetically) walking in the first place when you encountered the first indication of this Pokémon's existence?
I think the following extreme case demonstrates that it can in principle make a difference. Suppose you start from the origin, and follow an Archimedean spiral (the one maintaining constant distance between the path on two consecutive turns) with spacing x. Now, when you encounter the outside of the Pokémon's 200m-radius circle of detection, how might that circle be oriented relative to you?
The answer, surely, is that the smaller x is, the closer to almost certain it is that you have barely grazed the very edge of the circle of detection. Because if x = 1 metre, for example, then there's no way your path could be aimed at the very centre of the circle on iteration n, unless it had encountered the edge of the circle 199 turns of the spiral before that.
So if you're following a path of that form, then surely it's pretty much guaranteed that as soon as you catch the first inkling of a Pokémon's existence, you are moving more or less tangent to the edge of its circle, so to get close to the centre of the circle as efficiently as possible you should immediately turn 90° and head directly outwards from the origin.
But if instead you were following some totally different path, say a randomly chosen doubly infinite straight line in the plane (that is, with angle uniformly chosen and perpendicular distance to the origin chosen according to some concrete probability distribution which is uniform enough over the range in question), the answer will be very different and you'll have at least some chance of happening to be aimed somewhere near the centre of the circle.
In between those two extremes, following a path of the form ‘walk around town along whatever collection of well-worn routes make up your normal daily routine, plus a few specifically Pokémon-oriented excursions as and when you feel like it’, it's surely anybody's guess.
no subject
Date: 2016-08-24 02:37 pm (UTC)In particular: along what shape of path were you (hypothetically) walking in the first place when you encountered the first indication of this Pokémon's existence?
I think the following extreme case demonstrates that it can in principle make a difference. Suppose you start from the origin, and follow an Archimedean spiral (the one maintaining constant distance between the path on two consecutive turns) with spacing x. Now, when you encounter the outside of the Pokémon's 200m-radius circle of detection, how might that circle be oriented relative to you?
The answer, surely, is that the smaller x is, the closer to almost certain it is that you have barely grazed the very edge of the circle of detection. Because if x = 1 metre, for example, then there's no way your path could be aimed at the very centre of the circle on iteration n, unless it had encountered the edge of the circle 199 turns of the spiral before that.
So if you're following a path of that form, then surely it's pretty much guaranteed that as soon as you catch the first inkling of a Pokémon's existence, you are moving more or less tangent to the edge of its circle, so to get close to the centre of the circle as efficiently as possible you should immediately turn 90° and head directly outwards from the origin.
But if instead you were following some totally different path, say a randomly chosen doubly infinite straight line in the plane (that is, with angle uniformly chosen and perpendicular distance to the origin chosen according to some concrete probability distribution which is uniform enough over the range in question), the answer will be very different and you'll have at least some chance of happening to be aimed somewhere near the centre of the circle.
In between those two extremes, following a path of the form ‘walk around town along whatever collection of well-worn routes make up your normal daily routine, plus a few specifically Pokémon-oriented excursions as and when you feel like it’, it's surely anybody's guess.