I'm trying to figure out what the curvature of the hyperbola given by y = 1/x is, and I already know the answer. Unfortunately, I'm getting a strange factor of x^3 in my answer.
y = 1/x
f'(x) = -1/(x^2)
f''(x) = 2/(x^3)
k = |2/(x^3)|/((1/(x^4) + 1)^(3/2))
= 2/(|t^3| * (1 + 1/(t^4))^(3/2)) (as x is the same as the parameter t, (x, y) = (t, 1/t))
and p = 1/t = 1/2 * (1 + 1/(t^4))^(3/2) * |t^3|
But, the curvature which leads to the correct evolute for the curve (which is what I'm really looking for) is simply 1/2(1 + 1/(t^4))^(3/2), without the |t^3|. I've been working on this for an hour, and I still can't figure out why it disappears. Thanks for any help.