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)?
I am guessing the best path would be the best segmented replica of a spiral. If so, that leaves the terms "segmented" and "spiral" to be better defined.
There are many forms of spirals and perhaps one key parameter (as I recall) governing their form is their angular attack as they rotate in: a relation between theta and d_r/d_theta. The best (longest) spiral for this puzzle would have the highest ds/dr, where s is the line integral and r is the distance.
As for segmentation: does one use uniformly sized segments or segments that are of a length to best trace the curvature of each portion of the spiral: (note, as defined, they may have an infinite number of wraps, though not infinite length.)
Perhaps there is a trade-off here where the segment length vs. the length achieved is maximized.
The above is very messy, so hopefully I am over-thinking the problem.
Edited on January 14, 2019, 4:29 pm
some floundering ideas
