Points A and B are on a plane surface, 1 mile apart. Suppose you must walk in a path consisting of N straight lines from point A to point B, such that at all times your (Euclidean) distance to point B is decreasing. What is the longest possible route length (as a function of N)?
(In reply to
But.... by Kenny M)
Perhaps it's more intuitive for biggest numbers of N, like N=10000.Ten thousands little segments are a chain that can imitate very well a curve and the curve gives the biggest lenght.
For N=10000 each segment would have a lenght of 0,01 mile and the total lenght would be 100 miles.
Practically, as the lenght is always the same you need only a very accurate compass. When in A you can measure the angle given by arccos sqr(9999/10000). This is 0.57296. Measure this angle from the perpendicular to A-B but versus B and walk 0.01miles. There you measure the angle given by arccos (sqr (9998/9999)), which is 0.57299 and change direction that little always versus B. Walk for another 0.01 miles and then measure arccos (sqr (9997/9998). Turn in direction to B that 0.57302 angle and walk for anther 0.01 miles...
You will notice that for the most part of the time you are quite far from B and close to A, but little by little the angles are increasing and the shape of the spiral path, which has a high density when you are close to point A is progresively decreasing and you are quickier approching point B.
But once you are at point B, you are probably going to think: perhaps time has come for a visit to the psychiatrist...
Edited on January 16, 2019, 4:22 am
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Posted by armando
on 2019-01-16 02:59:20 |
re: But....
