Construct the involute for starting from (0,0).

My solution so far:

Let

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?

Thank you.

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- Feb 14th 2008, 07:15 PMtttcomraderConstruct involute
Construct the involute for starting from (0,0).

My solution so far:

Let

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?

Thank you. - Feb 15th 2008, 01:36 AMmr fantastic
That's the arc length you're stuck with, I'm afraid. But did you really expect the equations defining the involute to be simple?

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. - Feb 15th 2008, 02:23 AMearboth
If I understand and translate the expression "involute" correctly this curve is defined (in your question) as the locus of all intersection points of the tangents to the parabola and the normal to the tangents passing through the origin.

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:

and

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.