# Curvature of a Helix

1. Sep 30, 2008

### LRP0790

1. The problem statement, all variables and given/known data

Find the curvature of a helix given by the parametric equation r(t)=<acost, asint, bt> where a and b are real numbers

2. Relevant equations

I know k=|T'(t)/r'(t)|

3. The attempt at a solution

and I believe the answer to be k=b/(a2+b2)1/2, I just don't know how to get there

2. Sep 30, 2008

### HallsofIvy

Staff Emeritus
First step, write the formula correctly! You can't divide vectors!
Did you mean k= |T'(t)|/|r'(t)|?

If so then if r= <a cos t, a sin t, bt>, r'= <-a sin t, a cos t, b> and it's length is $|r'|= \sqrt{a^2 sin^2 t+ a^2 sin^2 t+ b^2}= \sqrt{a^2+ b^2}$, a constant. That means that T, the unit tangent vector is
$$T= \frac{1}{\sqrt{a^2+ b^2}}<-a sin t, a cos t, b>$$

That's easy to differentiate with respect to t (since that whole first fraction is a constant). Do that and take the length of |T'|. Divide by the length of r' which I've already given you.