Function on a 3 dimensional Cartesian plane that represents a helix?

Question

What I mean is that is that are functions similar to a DNA strand able to be graphed on a 3 dimensional coordinate plane, through something like polar coordinates? Thanks in advance!

Answer 1

I am assuming the helix has $z$-axis as its axis.

You can imagine:

1) The projection of any point on $x$-axis lies on a circle: This gives $r\cos\phi=R$

2) Value of $\theta$ is periodic wrt z: This gives $\theta=2\pi\cos\frac{2\pi z}{P}$

Here $P$ and $R$ are pitch and radius respectively.

We know $z=r\sin\phi$

Thus the helix is specified using 2 equations:

$$r\cos\phi=R;\theta=2\pi\cos\frac{2\pi r\sin\phi}{P}$$

Answer 2

You can represent a helix with radius r with the following parametric equations:

$$x=r \cos(t)$$ $$y=r\sin(t)$$ $$z=ct$$ for $t\in[0,2\pi]$. Additionally, $2\pi c$ gives the vertical distance between the loops of the helix. In terms of vector calculus, this could also be written as $$p(t)=(r\cos(t),r\sin(t),ct)$$ where t could represent time.