1
$\begingroup$

I am trying to find the parametric equation of a sine curve, which oscillates around a circle as it's $x$-axis. I have done preliminary approximations using Epicycloid parametric equations for the top portion of the wave and Hypocycloid parametric equations for the bottom portion while using the same exact same radii for the big and small circles for each case, respectively.

This actually worked pretty well, but I was wondering whether a sine curve, or something of the like, can be made in the same manner, since it is a little bit different. If you have any ideas or suggestions, it would be much appreciated.

References

Hypocycloid: $$x = (a-b)\cos \phi + b\cos \left( \frac{(a-b)}{b}\phi \right)$$ $$y = (a-b)\sin \phi - b\sin \left( \frac{(a-b)}{b}\phi \right)$$

Epicycloid: $$x = (a+b)\cos \phi - b\cos \left( \frac{(a+b)}{b}\phi \right)$$ $$y = (a+b)\sin \phi - b\sin \left( \frac{(a+b)}{b}\phi \right)$$

$\endgroup$
1
$\begingroup$

A more straight forward curve parametrized in polar coordinates:

\begin{align*} r &= a+b\sin \frac{b\theta}{a} \\ \begin{pmatrix} x \\ y \end{pmatrix} &= \begin{pmatrix} (a+b\sin \frac{b\theta}{a}) \cos \theta \\ (a+b\sin \frac{b\theta}{a}) \sin \theta \end{pmatrix} \\ \end{align*}

which doubles your frequency and halves your amplitude.

Replacing $\displaystyle \frac{b}{a}$ by $n$ yields any frequencies.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.