Curvature of a rectangular hyperbola

In summary, the hyperbola y = 1/x in the first quadrant has a parametric definition of (x, y) = (t, 1/t), t>0. The corresponding parametric form of its evolute is p = 1/t = 1/2 * (1 + 1/(t^4))^(3/2) * |t^3|. The curvature formula is |f''(x)|/((1+(f'(x))^2)^(3/2)), and the correct curvature for the evolute is 1/2(1 + 1/(t^4))^(3/2), without the |t^3|.
  • #1
StarWrecker
2
0

Homework Statement


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

Homework Equations



Curvature formula:

|f''(x)|/((1+(f'(x))^2)^(3/2))

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.
 
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  • #2
Hi,StarWrecker,I am 100 percent sure "p = 1/t = 1/2 * (1 + 1/(t^4))^(3/2) * |t^3|"
is correct.
And I am also sure that p(t) leads to the correct evolute for the curve.
Check your answer again and again,you will find out your errors on computation.
 
  • #3
Yeah, I made a mistake in my later calculations for the evolute that I ironed out after trying my solution for the curvature again. Thanks.
 

1. What is the curvature of a rectangular hyperbola?

The curvature of a rectangular hyperbola is a measure of how much the curve deviates from being a straight line at a particular point. It is determined by the rate at which the direction of the curve changes as you move along it.

2. How is the curvature of a rectangular hyperbola calculated?

The curvature of a rectangular hyperbola is calculated using the formula: κ = |dy/dx| / (1 + (dy/dx)^2)^(3/2). This involves finding the first and second derivatives of the equation of the hyperbola and plugging them into the formula.

3. What does a positive or negative curvature indicate for a rectangular hyperbola?

A positive curvature indicates that the curve is concave, meaning it bends towards the inside, while a negative curvature indicates that the curve is convex, meaning it bends towards the outside. In the case of a rectangular hyperbola, a positive curvature would mean that the hyperbola opens upwards and a negative curvature would mean that it opens downwards.

4. Can the curvature of a rectangular hyperbola be constant?

No, the curvature of a rectangular hyperbola is not constant. It varies at different points along the curve. However, in certain special cases, such as when the hyperbola is a circle, the curvature can be constant.

5. How does the curvature of a rectangular hyperbola relate to its asymptotes?

The curvature of a rectangular hyperbola is inversely proportional to the distance between its asymptotes. This means that as the curvature increases, the distance between the asymptotes decreases, and vice versa. This relationship is important in understanding the overall shape and behavior of rectangular hyperbolas.

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