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

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

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):

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

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

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

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

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

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

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

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):

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

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

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

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

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

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

Dear Ackbeet,

Thank you so much. So we cannot express a space curve by a single cartesian equation is'nt? I think so too. But the strange thing is I didnt find it mentioned anywhere.

4. So we cannot express a space curve by a single cartesian equation...?
No, I don't think you can. And that's merely because of the fact that if you think about the process of solving equations, one equation will generally allow you to eliminate one variable. 3D space obviously has 3 variables, x, y, and z. But a space curve has only one degree of freedom, the parameter t or whatever you want to call it. To get down to one variable from three requires two equations. So there you go.

This is something that mathematicians may or may not mention. The physicists definitely talk about the number of degrees of freedom (variables) all the time. Maybe that's because physicists are sometimes more interested in providing the actual solution, instead of just proving that a solution exists.

Cheers.

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standard equation of a helix

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