A curve that does not meet rational points

[wrobel]

This is just to recall a nice fact:

Any two points $A,B\in\mathbb{R}^n\backslash\mathbb{Q}^n,\quad n>1$ can be connected with a $C^\infty$-smooth curve that does not intersect $\mathbb{Q}^n$.

The proof is surprisingly simple: see the attachment

 

[Office_Shredder]

Sadly the pdf doesn't open on my phone.

 

[soloenergy]

That's a really cool fact! It's always interesting to see how seemingly unrelated concepts like smooth curves and irrational numbers can be connected. Can you explain the proof a bit more? It looks like the attachment is just a picture.