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Let say we have a metric space $(M,d)$ (in particular, we will require it to be a length metric space). If an airplane in that space that moves at speed of 1, then it can get from point $a$ to point $b$ in $d(a,b)$ time.

Now let us suppose we add wind to this space. By wind, I mean we assign to each point $x$ a vector $w(x)$.

When the airplane is at point $a$ moving in direction $v$ (which is a unit vector), it "moves" according to $v + w(a)$. In affect, the wind is blowing against it.

We know define a quasimetric (M, d') as follows. $d'(a,b)$ is defined as the shortest possible amount of time it would take the airplane to get from point $a$ to point $b$ (or $\infty$ if the airplane can't get to $a$ from $b$).

Is there a name for this concept of turning a metric into a quasimetric by adding "wind" to the space?

Examples:

  • If we define $w(x)=\vec 0$ for all $x$, then $(M,d) = (M,d')$.
  • If we take a circle, and define $w(x)$ as a unit vector going clockwise, we get the quasimetric defined in the first paragraph in this answer (except the distances are halved).
Christopher King
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  • Perhaps geometric group action, if you think of the wind as describing a transformation of the space sending points on the space to other points on the space. Is the airplane blown only by wind, or can it move under its own power? – user326210 Jan 25 '18 at 07:36
  • I'm not sure what this means, but perhaps it's a typo: $v+w(a)$ instead of $v+d(a)$? – user326210 Jan 25 '18 at 07:37
  • @user326210 it can move under it's own power. It has a constant air speed of 1. – Christopher King Jan 25 '18 at 10:01

2 Answers2

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This sort of thing is modelled using Finsler metrics.

If you like wind, then Zermelo navigation metrics is what you want.

Mariano Suárez-Álvarez
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If you deal with a riemannian manifold, let $g_p $ be the inner product in $T_pM$ (a positive definite matrix whose entries vary smoothly) . Given a scalar smooth function $w(p) > 0$, consider the new inner product $h_p= w(p) g_p$ (you multiply the entire matrix by a number).

Recall that the length of a path $\gamma: (-1,1) \to M$ is $$\int_{-1}^{1} \gamma'(t) ^T g_{\gamma(t)} \gamma'(t) dt $$ With the modified version of metrics, you have an extra factor $w(\gamma(t)) $. If you started with an airplane having velocity 1 in the old metrics, you have $\gamma'(t) ^T g_{\gamma(t)} \gamma'(t)=1$; so with the extra factor you get that the new distance is $\int w(\gamma(t)) dt $.This means that in points with "high" wind you will have high $w(p) $ (which in some sense measure how much it is difficult to go over a point).

This is not exactly what you wanted because this change is linear, but it can be a good approximation when the wind is little and it can be assumed to be proportional to the velocity.

Andrea Marino
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