Parametric equation for a helix winding around an arbitrary line

Let l(t)=p+vt where l,p,v are vectors. Say p=r_{0}=<1,1,1> and v=<1,2,3> and the helix has radius 2.

So I have l(t)=<1+t,1+2t,1+3t> and that the equation of a helix would be <2cost,2sint,t>

How do I find the x(t),y(t), and z(t)?

I also found that sin(theta)=2/r(t), but I know that theta will get smaller as t goes on.

I'm studying for my multivariable calc final and this one is just stumping me! It feels like I should know the answer to this.

Thank you!

Re: Parametric equation for a helix winding around an arbitrary line

I think I got it

Would you simply add the parametric equation of the helix to the parametric equation of the line?

l(t)=<1+t,1+2t,1+3t> and h(t)=<cost,sint,t>

so x(t)=1+t+cost, y(t)=1+2t+sint, z(t)=1+3t+t=1+4t

I believe this works. Someone correct me if I'm wrong.

1 Attachment(s)

Re: Parametric equation for a helix winding around an arbitrary line

Hi,

No, what you are suggesting is to translate the given curve, but with a varying t in the translation. You don't get a helix. Here's a drawing of what you get:

Attachment 28259

See the following post for a correct solution.

2 Attachment(s)

Re: Parametric equation for a helix winding around an arbitrary line

Hi,

First, I've attached a discussion about how to change coordinates to produce the desired helix. The second attachment shows the "standard" helix as a dotted curve, the line (1,1,1) + t(1,2,3) and the helix which wraps around this line:

Attachment 28260

Attachment 28261