Sinewave riding on sinewave help

Question

Consider this image

Top one is Cos(t), I know that. What is the equation for the second one? (sinewave within sinewave) And how would I get to the third one and then on n-amount of recursion?

Mind boggling for me!

Answer 1

Well, given a parameterized curve $f$, we want to create a new curve $g$ which is the same as $f$, but with some oscillations added. In particular, we want the oscillations in $g$ to always be perpendicular to motion of $f$. So, in vectors, if we let $\hat{f}(t)$ be normal (perpendicular vector) to $f$ at time $t$ we could write: $$g(t)=f(t)+a\sin(bt)\hat{f}(t)$$ where $a$ is the amplitude of the new wave and $b$ is the frequency - which are parameters you can play with to your heart's content. Note that you could easily define $\hat{f}$ as a 90 degree rotation and normalization (i.e. divide by magnitude) of $f'(t)$, the derivative. And, if you start this process with $f(t)=(t,0)$ - a line - and repeatedly apply the transformation that takes $f$ to $g$, you will get nested waves like this. However, this expression gets ugly very quickly if you want a closed form; for instance, with 2 nested waves for certain parameters that I thought seemed reasonable, one gets the expression

$$\left\{x-\frac{\sin (20 x) \cos (x)}{3 \sqrt{\cos ^2(x)+1}},\sin (x)+\frac{\sin (20 x)}{3 \sqrt{\cos ^2(x)+1}}\right\}$$ This is ugly. You don't want to see the expression for 3 nested waves.

Note that a similar method would work in 3 dimensions, but since the set of vectors normal to a 3D curve two dimensional, it's trickier.