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Is there a specific function that takes the real number line $\mathbb{R}$ and converts it into a helix?

guego
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2 Answers2

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There is:

$$\begin{array}{rcl}\mathbb{R}&\longrightarrow&\mathbb{R}^3\\t&\longmapsto&(\sin t,\cos t,t)\end{array}$$

Jared
  • 31,451
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Would $$ t\mapsto(a\cos t, a\sin t, bt) $$ fit the bill? Its image is a helix. The parameters $a$ and $b$ can be any non-zero constants. They determine the rise/rotation and handedness of the helix.

Jyrki Lahtonen
  • 133,153
  • We can apply this transformation to also get a torus? – guego Jun 12 '13 at 18:26
  • Torus is a surface, so you need two parameters to describe it. Use something like $$(u,v)\mapsto((a+b\cos u)\cos v, (a+b\cos u)\sin v,b\sin u)$$ and let both $u,v$ range over $[0,2\pi)$. Here $a$ is the radius of the circle "in the middle of the tube", and $b$ is the radius of the tube. You want $0<b<a$. – Jyrki Lahtonen Jun 12 '13 at 18:36