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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.

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kobu-kuro
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  • What is a circle with infinite length? Do you mean something like the Koch snowflake, which is a curve with (in a precise sense) infinite length? – Noah Schweber Mar 09 '16 at 02:44
  • All maps of the circle to the plane are homotopic. Some of them have zero length, some of them have positive, finite length, some of them have infinite length. So yes, a homotopy as you ask can exist. – Lee Mosher Mar 09 '16 at 03:28
  • You have completely effaced the previous question here and put a new question in its place. Do not do that, it has the very undesirable effect of making the comments people have already written entirely nonsensical. If you want to change your question, ask a new one instead. – Lee Mosher Mar 09 '16 at 13:16
  • @LeeMosher I am very sorry. I noticed the previous question was not I had meant. I won't dot that again in this site. – kobu-kuro Mar 10 '16 at 05:06

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