It reminds me of Kohonen maps,

http://en.wikipedia.org/wiki/Self-organizing_map#Learning_algorithm

or what is done in Captchas

http://en.wikipedia.org/wiki/Captcha

For example, choose some random points in

the plane (here A,B,C). To deform a path or

a pixel, consider the distances from a point (D)

on the path to each of the random points,

here A-D, B-D and C-D. Apply a function f on

it: f(A-D), f(B-D), f(C-D). This gives displacement

vectors. Here the arrows. Add them to the old

position to obtain the new point.

You may want f to be nonlinear, e.g. exp(x).

Regards,

Joachim

I'm not looking for functions to deform specific trajectories. What I want is to be able to use a trajectory-based constraint to define a 2D deformation of a planar shape, not a single trajectory. There are plenty of techniques in the animation/computer graphics communities, but they're focused on constraining the deformation based on things like smoothness during the transition or realism of the deformed shape (so when you move a hoof on an animated horse it still looks like a horse, not like a horse with a broken leg). I haven't found any techniques that constrain the deformation based on properties of trajectories through the space instead.

An Isometry will do the job. It preserves the lengths of curves and thus longer curves after the transformation will still be longer than the sorter ones.

Ioannis' response is correct. If you want the transformation to be defined everywhere and for this ordering preservation to hold for every pair of curves then you must have a combination of an isometry or global scaling. In other words, your only options are translation, scaling and rotation. Such transformations are generally called "similarity transforms", since they would also map polygons to similar polygons. Specific tools for computing these are available in many image registration toolkits, including ITK.

The second option you describe is unfortunately kind of hard to formalize. In particular, taking means over an infinite dimensional space of paths has its complications. However, if you are looking for a "softened" deformation model without such a harsh constraint as the finite-dimensional similarity transforms, then there are many useful models that use metrics with interpretations that sound similar to this. In particular the diffeomorphic image registration algorithms used in computational anatomy and medical image analysis would be of interest (search terms: LDDMM, b-splines, demons).

Hope this is of some use.