This post is some musings in relation to the New York Times NumberPlay: “Joe Fendel’s Long Walk Home”.
The constraint that the distance to the destination confines the paths. Without loss of generality let A be (-1,0) and B (0,0). Therefore, the constraint is satisfied if the norm of points of the path are strictly decreasing. I made a Mathematica simulation using
LocatorPane. The first panel is the path. The second panel is the slope and the last panel is the distance. It divides the segment into 10 segments but could be further divided and consequently distance increased.
If I get time I will think about more rigorously.
An interactive CDF is here.