- #1

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## Main Question or Discussion Point

Supposing I have a curve, [itex]\vec{f}\left(s,t\right)[/itex] that lives in [itex]\Re^{3}[/itex] and is deforming over time but never intersecting itself (s is the curve parameter and t is time). I would like to extend the deformation of the curve to the entire space around the curve, creating a transformation of the whole space [itex]\vec{g}\left(\vec{x},t\right)[/itex] that is continuous and doesn't overlap itself.

It would be fine to restrict the curve to a series of connected straight line segments, if that would make it easier.

If I can construct such a function, g, I believe I can use it to create a very general method of solving "tanglement puzzles", where the object is to remove a flexible loop of string from a metal contraption.

Here are some pictures that describe what I am taking about.

Here is the function I start with:

http://img381.imageshack.us/img381/1968/pathcurvingsmww1.png [Broken]

and here is the function I want to construct:

http://img162.imageshack.us/img162/7487/pathcurvingcoordssmcz2.png [Broken]

And here are a couple examples of "remove the string" tanglement puzzles:

http://www.puzzles.ca/puzzle_data_3/xastroknot_l.jpg

It would be fine to restrict the curve to a series of connected straight line segments, if that would make it easier.

If I can construct such a function, g, I believe I can use it to create a very general method of solving "tanglement puzzles", where the object is to remove a flexible loop of string from a metal contraption.

Here are some pictures that describe what I am taking about.

Here is the function I start with:

http://img381.imageshack.us/img381/1968/pathcurvingsmww1.png [Broken]

and here is the function I want to construct:

http://img162.imageshack.us/img162/7487/pathcurvingcoordssmcz2.png [Broken]

And here are a couple examples of "remove the string" tanglement puzzles:

http://www.puzzles.ca/puzzle_data_3/xastroknot_l.jpg

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