Lipschitz embeddings of algebraic curves

We show that inner Lipschitz classification of real algebraic curves is equivalent to outer Lipschitz classification in $\mathbb{R}^3$ of real algebraic curves with LNE connected components.

Moreover, for every real affine algebraic curve we find its LNE model which is real birational to it.

Lastly, we show that this claim does not hold in the complex category.