Help creating a specific smooth curve

[StillNihilist]

Note: If this is the wrong sub-forum for this question please move it. I was not sure if this question should go in the General section or not.

Question:
I want to create a smooth non-piecewise curve in ℝ$^{3}$ (3-space) such that it's intersection with the xy-plane consists of the integer coordinates Z$^{2}$. If this is impossible for the entire xy-plane, I'd like to be able to create such a curve with that intersection property for some arbitrary rectangular sub-section of the xy-plane.

I can visualize such a curve as being like a thread weaving up and down through the integer coordinates, either spiraling out from some initial point or maybe weaving back and forth across the grid. Furthermore, it's rather easy to do this for only 2-dimensions, for example, the curve c(t) = <t, sin($\pi$t)>, works for two dimensions. However I can't find an algebraic representation of such a curve in 3-dimensions.

Any help is extremely appreciated. Thank you for your time and consideration. I hope you have a great day.

 

[LCKurtz]

Not sure how much help this will be, just a couple of thoughts. If you take the two variable function $f(x,y) = \sin(\pi x)\sin(\pi y)$, that surface is zero along lines $x=n$ and $y=m$ (integers), and nowhere else. Now think about the curve $|x|+|y| = k$ for some natural number $k$. That will pass through integer coordinates forming a diagonal square. If you restrict the surface $z=f(x,y)$ to that square domain, you will get such a curve. It may not be smooth, and it doesn't solve your problem, but here's what it looks like for $k=3$: