Construct the involute for starting from (0,0).
My solution so far:
I use the formula where
I find that
But in solving s(t), I have: , but I can't see a way to simple this, am I doing this right?
Things aren't that bad .... work with what you've got.
By the way, the integral for the arc length can be expressed in terms of the inverse hyperbolic sine function. Because I'm lazy, I'll just suggest you fire up the ol' Wolfram Integrator.
To put your mind at rest, you might want to look at this.
Let the tangent point of the parabola.
Then the equation of the tangent to the parabola in T is:
. The equation of the normal to this tangent through the origin is:
Then the intersection point has the coordinates:
This is simultaneously the parametrized equation of the involute. There exists a asymptote because
After a few steps of transformation (elimination of the parameter t, etc) you'll get the equation:
which describes the involute.