3
$\begingroup$

(This question was originally asked on Mathematics Stack Exchange, and sat there for several weeks with low views and no answers.)

Let $\phi$ and $\gamma$ be rectifiable curves in the same length space. Declare the length of $\phi$ relative to $\gamma$ ('relative length') as $ {\mathrm{Length}(\phi)} / {\mathrm{Length}(\gamma)} $.

Now let $f : \mathbb{R}^2 \rightarrow \mathbb{R}^n$ and $M$ be a path-connected subset of $\mathbb{R}^2$ such that $f$ is everywhere differentiable on $M$. In general, the length of $f\circ\phi$ relative to $f\circ\gamma$ will be different to the length of $\phi$ relative to $\gamma$.

What conditions can we place on $f$, $M$, or the curves so that the change in relative length is bounded?

If it helps, I am specifically interested in the case where $\phi$ and $\gamma$ are made up of a finite number of line segments, $M$ is a rectangle in the plane, and $f$ is a diffeomorphism that preserves geodesics.

Why I am asking

My question comes from research I am doing on short-path planning across 2-dimensional surfaces. Earlier work has established that for a given shortest path in the plane, there exists a path with the same start and goal that is (potentially) cheaper to find and has an upper bound on the difference in relative length. That is in the plane, one can find a short-enough path quickly versus a shortest path slowly.

I am seeking to use this work by transforming queries about short paths across a surface into queries about short paths in the plane, and then transforming the results back onto the surface. When transforming to the surface, there could well be a change in the difference in relative lengths. I would like to know if there is an upper bound on the change.

What I have tried

If $f$ is an affine map then its Jacobian $J$ is constant. Moreover if $\sigma_1$ and $\sigma_2$ are the singular values of $J$ with $\sigma_1 \leq \sigma_2$ then line segments suffer a change in relative length of at most $\sigma_2 / \sigma_1$ :

  • If the line segments are parallel then there is no change in relative length.

  • But if they are perpendicular then they suffer the maximum change in relative length.

(See also this beautiful article on the singular value decomposition at the AMS website.)

My intuition is that if $M$ is compact (so $J$ is bounded) then we can loosen the conditions on $f$ to being 'it stretches by possibly different magnitudes in roughly the same direction'. This came from thinking about $f$ acting on a triangle ABC and thinking about relative length of path ABC vs path AC:

  • If I am restricted to actions that stretch AB and BC away from AC 'in roughly the same direction' then I think I can build an upper bound on the change in relative length by considering the biggest stretch

  • But if I can reverse the direction of stretching then I can make the change in relative length arbitrarily large by stretching AB one way and then the other in rapid succession, 'squiggling back and forth' as high-frequency zigzags across the base course ABC.

Any and all suggestions most appreciated, thanks very much.

$\endgroup$

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.