Is the Involute of a Catenary a Reparametrized Tractrix?

[kezman]

I need your help!.
Show that the involute of the catenary parameterized by arc length is a reparametrization of tractrix. (use the change of parameter 1/cosh(t)=sin(r))

The involute of the catenary (with arc length param.) gave me the following result:

(arcsinh(s), cosh(arcsinh(s))) + (s_o - s) (1\sqrt(s^2 + 1)), (s\sqrt(s^2 + 1)))

when I use the change of parameter s = sinh(t) I have this:

(t,cosh(t)) + (sinh(t_0) - sinh(t)) (1\(cosh(t)) , tgh(t))

I don't know how to continue with what its proposed 1/cosh(t)=sin(r) or if everything I did is worong .
Thanks.

 

[jvicens]

Hello there,

Thank you for reaching out for help. I am a scientist and I would be happy to assist you with this problem.

First, let's start by defining the involute of the catenary. The involute of a curve is the curve traced by the end of a string as it unwinds from the curve. In this case, the curve we are interested in is the catenary, which is the curve formed by a hanging chain or cable under its own weight.

Now, let's consider the catenary parameterized by arc length. This means that the arc length along the catenary curve is used as the parameter instead of the traditional x and y coordinates. Using this parameterization, we can write the catenary curve as (arcsinh(s), cosh(arcsinh(s))).

Next, we need to find the involute of this catenary curve. To do this, we can use the following formula: (x(s), y(s)) + (s_o - s) (y'(s), -x'(s)), where (x(s), y(s)) is the original curve and (x'(s), y'(s)) is its derivative.

Applying this formula to our catenary curve, we get (arcsinh(s), cosh(arcsinh(s))) + (s_o - s) (1/sqrt(s^2 + 1), s/sqrt(s^2 + 1)).

Now, we can use the change of parameter 1/cosh(t) = sin(r). This means that t = arcsinh(s) and r = cosh(arcsinh(s)). Substituting these values into our formula for the involute, we get (t, r) + (sinh(t_0) - sinh(t)) (1/cosh(t), tanh(t)).

This is the same result that you got, but using a different notation for the parameters. We can rewrite this in terms of s by substituting t = arcsinh(s) and r = cosh(arcsinh(s)). This gives us (arcsinh(s), cosh(arcsinh(s))) + (sinh(arcsinh(s_0)) - sinh(arcsinh(s))) (1/cosh(arcsinh(s)), tanh(arcsinh(s))).

Finally, we can simplify this to (arcsinh(s), cosh(arcsinh(s))) + (s_o - s) (1