mmmm today i will read an introduction on wave fronts

> everything is about the equidistant


there are several examples of Legendrian maps but the equidistant is like the easiest to parametrize imo

it makes sense it is the equidistant too

Basically, if you have a hypersurface \(H \subset \mathbb{R}^n\) and a point \(q \in \mathbb{R}^n\) outside \(H\), you can always consider the Gaussian field \(N\colon \mathbb{R}^n \longrightarrow T_1\mathbb{R}^n\) (the \(1\) means it's been normalized) and compute the sets

\(\mbox{Eq}(q,t)=\{q+tN_p: p \in H\}\)


if you draw that out, you will note that it looks like the ripples emanating from the point \(q\). For example, here you have some images of what happens if you consider the equidistant of an ellipse:

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I like to explain it like this: for \(n=2\), \(H\) (and everything inside) is a pond where you throw a rock at the point \(q\). The equidistants are then the ripples that happen on the surface of the water.

Of course it doesn't really make sense to consider \(q \in H\) or in the outside of the region (no water). But afaik the interesting thing with equidistants happens when the hypersurface is close and bounded, so that we have a well-defined "outside" and "inside", and the point is inside.

Equidistants come in any context where you have a wave travelling in space, so you could also consider sound waves or rays of light.

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