MHB How Is the Unit Normal Derived in an Epicycloid Equation?

[bugatti79]

Hi Folks,

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

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

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

2) Is the bottom just based on simple trig identities? Wolfram didnt simply the denominator
simplify '('sin A -m sin'('A'+'B')'')''^'2 - Wolfram|Alpha

 

[poissonspot]

Hi bugatti79,

The $r+\rho$ term cancels upon dividing $N_x$ by the length $|N|=\sqrt{|N_x|^2+|N_y|^2}$.

As for the second part, Pythagorean and sum-difference identities establish the equality:
$({\sin(\theta)-{m}{\sin(\theta+\psi)}})^2+({\cos(\theta)-{m}{\cos(\theta+\psi)}})^2=1-2{m}{\cos(\psi)}+m^2$