Deriving the Unit Normal for an Epicycloid: Understanding Equation 3.2.29

[bugatti79]

Hi Folks,

I got stuck towards the end where it ask to derive the unit normal (eqn 3.2.29 I don't know how they arrived at n_x. I have looked at trig identities...and I have assumed the following

n_x=\frac{N_x}{|N_x|}

I don't see the (r+p) term anywhere in neither the top nor bottom.

PS: I have posted this in MHB on Tues but no response.
http://mathhelpboards.com/calculus-10/unit-normal-epicycloid-16922.html

Thanks

 

[fzero]

The norm of the normal vector is $|\mathbf{N}| = \sqrt{ N_x^2 + N_y^2 }$. The unit normal is $\mathbf{n} = \mathbf{N}/ |\mathbf{N}|$, so
$$ n_x = \frac{ N_x}{\sqrt{N_x^2 + N_y^2}}.$$
In particular, there are common factors of $r+\rho$ in the numerator and denominator that cancel out. The rest of the simplification of the denominator can be accomplished with some trig identities.

 

[bugatti79]

Ok, great thanks. I should work it out now.