Parametric Equation of sine wave along a helix curve

Question

I would like to model a braided hose like shown here: https://grabcad.com/tutorials/tutorial-how-to-model-a-braided-hose-in-solidworks

I found these equations for sine along a helix: Parametric Equation of sine wave helically wrapped around a cylinder

I was wondering how the equations provided here would change if the sine wave has to follow a helical path, but in my case the wave is perpendicular to the cylinder surface instead of being in the axial direction.

Really appreciate the help.

Answer 1

The projection onto horizontal plane $z=0$ of the 3D curve you want to obtain can be given the following polar representation :

$$r(\theta)=R+r \cos(k \theta) \ \iff \ \begin{cases}x(\theta)&=&(R+r\cos(k \theta))\cos \theta\\y(\theta)&=&(R+r\cos(k \theta))\sin \theta\end{cases}\tag{1}$$

(Consider different values $k=5,10,20...$).

Retrieving the 3D curve out of this 2D curve if obtained through a natural "lifting", i.e., by adding to the 2 equations in (1) the third equation:

$$z(\theta)=a \theta$$

for a certain constant $a$.

Remark: a shift parameter $\phi$ can be added to the equations in (1) by taking $r(\theta)=(R+r \cos(k(\theta+ \phi)))$.