Let me start with an R^2 ambient isotopy J taking a straight line C1 to some C2.
An answer of other question implies that it can happen that you cannot define the length for C2.
[Answer](Two ambient isotopic curve segments, one has the length and the other does not )
Let me add some rules on the question and ask if the similar answer holds.
Let J be some R^2 ambiet isotopy taking the interval [1,2] on the X axis.
J: [0,1] * R^2 -> R^2
J is restricted so that all point will not change its x coordinate.
Then let me ask if it can happen that you can not define the length of J(1,[1,2]) or it has the infinite length.
In detail, I mean that J takes the interval [1,2] on the X axis to J(1,[1,2]).
Thank you in advance.