The power of backtracking and the confinement of length
We show that there is a point on a computable arc that does not belong to any computable rectifiable curve. We also show that there is a point on a computable rectifiable curve with computable length that does not belong to any computable arc.
This is a manuscript of an article published as McNicholl, Timothy. "The power of backtracking and the confinement of length." Proceedings of the American Mathematical Society 141, no. 3 (2013): 1041-1053, doi:10.1090/S0002-9939-2012-11385-1. Posted with permission.