Non embedded curve in $\mathbb{R}^3$

Question

What does it mean to say that a curve in $\mathbb{ R}^3$ is non-embedded ?

I think if it's lies on a plane , but i cant find any book who give this definition or some examples

thank you for your help

Answer 1

If you take a figure-8, it can be drawn smoothly, you get an immersion of the circle, but not an embedding; an embedding is required to be 1-to-1, but the lemniscate has a "crossing" in the middle.

Here's a parametric version; you can defined it for $0 \le t \le 2 \pi$, for instance: $$ x = \frac{\cos t}{1+\sin^2 t} \\ y = \frac{\sin t \cos t}{1+\sin^2 t} $$

An embedding it also required to be smooth, so that $$ t \mapsto (t, t^\frac{1}{3}, 0) $$ for instance, is not an embedding, although the image of that map is indeed also the image of an embedding, so you can't always tell that a map is a non-embedding by merely looking at its image. Finally, there are cases where looking at the image is enough, like $$ t \mapsto (t, t^\frac{2}{3}, 0). $$

This curve, at $t = 0$, has a "cusp", and no amount of reparameterizing will make it smooth.