StarWrecker
Sep10-09, 10:34 AM
1. The problem statement, all variables and given/known data
The hyperbola y = 1/x in the first quadrant can be given the parametric definition (x, y) = (t, 1/t), t>0.
Find the corresponding parametric form of its evolute, and sketch both curves in the region 0<x<10, 0<y<10
2. Relevant equations
Curvature formula:
|f''(x)|/((1+(f'(x))^2)^(3/2))
3. The attempt at a solution
I've worked through this in full, but there is a single term in my curvature that is throwing off my end result. My calculation of this curvature is as follows:
So
f'(x) = -1/(x^2)
f''(x) = 2/(x^3)
And therefore
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.
The hyperbola y = 1/x in the first quadrant can be given the parametric definition (x, y) = (t, 1/t), t>0.
Find the corresponding parametric form of its evolute, and sketch both curves in the region 0<x<10, 0<y<10
2. Relevant equations
Curvature formula:
|f''(x)|/((1+(f'(x))^2)^(3/2))
3. The attempt at a solution
I've worked through this in full, but there is a single term in my curvature that is throwing off my end result. My calculation of this curvature is as follows:
So
f'(x) = -1/(x^2)
f''(x) = 2/(x^3)
And therefore
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.