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Question:

I want to create a smooth non-piecewise curve in ℝ[itex]^{3}[/itex] (3-space) such that it's intersection with the xy-plane consists of the integer coordinates Z[itex]^{2}[/itex]. 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([itex]\pi[/itex]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.

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# Help creating a specific smooth curve

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