# Thread: Cartesian equation of a helix.

1. ## Cartesian equation of a helix.

Hi everyone,

I was wondering whether there is a cartesian equation for a helix. If so what is it? Also how do we convert a three dimensional parametric equation to its cartesian form? (such as the equation of the helix in parametric form)

2. Well, the parametric equation for a helix could be

$\mathbf{r}=\langle \cos(\theta),\sin(\theta),\theta \rangle.$

So, you could have the two equations (I think you'd have to have two cartesian equations to represent a space curve, simply because of the degrees of freedom involved):

$x=\cos(z), y=\sin(z).$

In general, if you have a helix, the axis of which is parallel to one of the three main axes, then I'd solve that equation for the parameter, and then plug back into the other two equations. For example, take the helix

$\mathbf{r}=\langle 5\cos(\theta),\theta-2,-5\sin(\theta) \rangle.$

I'd solve $y=\theta-2$ for $\theta,$ giving $\theta=y+2,$ and then plug into the other two equations, yielding

$x=5\cos(y+2), z=-5\sin(y+2).$

If the axis is not parallel to one of the three main axes, things get a lot dicier.

3. Originally Posted by Ackbeet
Well, the parametric equation for a helix could be

$\mathbf{r}=\langle \cos(\theta),\sin(\theta),\theta \rangle.$

So, you could have the two equations (I think you'd have to have two cartesian equations to represent a space curve, simply because of the degrees of freedom involved):

$x=\cos(z), y=\sin(z).$

In general, if you have a helix, the axis of which is parallel to one of the three main axes, then I'd solve that equation for the parameter, and then plug back into the other two equations. For example, take the helix

$\mathbf{r}=\langle 5\cos(\theta),\theta-2,-5\sin(\theta) \rangle.$

I'd solve $y=\theta-2$ for $\theta,$ giving $\theta=y+2,$ and then plug into the other two equations, yielding

$x=5\cos(y+2), z=-5\sin(y+2).$

If the axis is not parallel to one of the three main axes, things get a lot dicier.