mmmm today i will read an introduction on wave fronts

> everything is about the equidistant

$$noice$$

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. The social network of the future: No ads, no corporate surveillance, ethical design, and decentralization! Own your data with Mastodon!