Navigating the globe
December 28th, 2006 ~ Posted in: MathematicsWalk a mile east, a mile north, a mile west, and a mile south. Where are you, relative to where you started?
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6 Responses to “Navigating the globe”
December 28th, 2006 at 7:05 pm
Somewhere west of where you started? (the circles of equal latitude being smaller the higher you go).
December 28th, 2006 at 7:10 pm
Correction: depends on where you start from. Somewhere to the east if you start in the southern hemisphere, upto half a mile below the equator. Somewhere to the west if you start in that half mile or in the northern hemisphere.
Am I right this time?
December 28th, 2006 at 9:16 pm
That seems right, but there’s no substitute for grinding out the calculations… Not that I’ve done them
December 28th, 2006 at 9:26 pm
Hmm, seems like if you start close enough to the north pole (say 1/2 mi south of it), the answer isn’t anything nice. You can’t start at either of the poles, so it seems there are three zones: the southern and northern hemispheres (with the band you mentioned), and the region within 1/2 mi from the north pole.
December 29th, 2006 at 10:14 pm
Speaking of the region around a north pole, I guess there is a problem of interpretation as soon as you are less than 1 mi from the pole. Say you are 0.75 mi away. What does it mean to go 1 mi north? Which way do we walk after we reach the pole?
I was wondering if there is some ‘local’ equivalent of your question to which we can answer “westward in the northern hemisphere including the equator, eastward in the southern hemisphere, except at the poles”. One way would be replacing 1 mi in the question with some parameter x and looking at the limit of the direction of the final displacement as x tends to 0. Or is there a saner alternative?
December 29th, 2006 at 10:36 pm
That’s a nice idea, and a neat way to keep the solution intuitive.
If– as I was doing without reflection– we take the meaning of phrases like ‘go north 1 mi’ to be ’start traveling in the direction of increasing latitude at this point, and continue in the same direction until you have traveled 1 mile’, so that if you reached the north pole with some distance remaining, you’d continue, heading south, all bets are off. Maybe this forces you to do the plug and chug.
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