Equation of a 3D spiral

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I want to know if a 3D spiral, that looks like this:

can be approximated to any sort of geometric primitive that can be described with a known equation, like some sort of twisted cylinder I suppose.

Thanks

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3 Answers

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Equation for a helix: $$x(t) = R \cos t, \quad y(t) = R \sin(t), \quad z(t) = at.$$

If you actually want a surface, then use the above to write $$(x-x(z/a))^2+(y-y(z/a))^2 = r^2$$ or $$(x-R \cos(z/a))^2+(y-R \sin(z/a))^2 = r^2$$ where $r$ is the radius of the "tube" and $R$ is the winding radius. In your particular picture you will have $a >> 1$ and $r, R = o(1)$.

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These are the parametric equations of a corkscrew \begin{equation*} \begin{aligned} x&=a \cos(\theta)\\ y&=a \sin(\theta)\\ z&=a \theta\tan(\alpha) \\ \end{aligned} \end{equation*}

Where $a$ is constant, the radius of the cylinder, and $\alpha$ is the constant angle made by the tangent with the $x, y$ plane.

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The tube thickness can also be varied as a function of any desired parameter ( radius, axial length, theta, arc length along tube etc.), the following image is that of of a helix on a cone. Code line is in Mathematica.

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