# Equation of a 3D spiral

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

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 $r, R = o(1)$.

• Does he just want a line or a shape with volume? May 21, 2014 at 23:02
• I have him both. The first is just the line. The second is surface thought of as a circle whose position moves depending on z. May 21, 2014 at 23:03
• Thanks, yep - wanted to be able to dictate the width of the "tube". May 21, 2014 at 23:38
• The implicit equation is bad because a helix is twisted so the circular section is not parallel to the $xy$-plane. But for helicoid, we can have an implicit equation: $x\sin \frac{z}{b}=y\cos \frac{z}{b}$ and for torus $(x^{2}+y^{2}+z^{2}+a^{2}-b^{2})^{2}=4a(x^{2}+y^{2})$. Jan 31, 2016 at 19:23
• I agree with @NgChungTak. The circular cross-sections should IMO be orthogonal to the curve - not parallel to the $xy$-plane. If $a$ is not large, then the distortion becomes clear (props for observing the need to have $a>>1$ though). See my old answer. Apr 3, 2018 at 5:46

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.

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.