First of all, I think the following question is more of a Mathematical question than physics question, so I am asking in math stackexchange.
Assume that a particle is rotating in $x$-$y$ plane about an axis ( call it $z$-axis) perpendicular to this plane. At any time, $t$, the $x$ and $y$ coordinate of the particle is written as:
\begin{equation} x = a \sin(\omega t)\\ y = a \cos(\omega t) \end{equation}
and the velocity of the particle is written as:
\begin{equation} \dot{x} = a\omega \cos(\omega t)\\ \dot{y} = -a\omega \sin(\omega t) \end{equation}
So, in $x$-$y$ plane the motion of the particle is represented by four parameters namely, $\dot{x},\dot{y},\omega, a$. I want to project this motion in $x$-$z$ plane (or $y$-$z$) plane to see if the motion of the particle can be represented by less number of parameters. Can anyone provide equation for the projection of this circular motion in the perpendicular plane.
In the $x,z$ plane it will simply be $$ \left\{ \matrix{ x = a\sin \left( {\omega t} \right) \hfill \cr z = 0 \hfill \cr} \right. $$
For other directions, put it vectorially $$ {\bf r}(t) = \left( {x(t),y(t),z(t)} \right) = \left( {a\sin \left( {\omega t} \right),a\cos \left( {\omega t} \right),0} \right) = a\left( {\sin \left( {\omega t} \right),\cos \left( {\omega t} \right),0} \right) $$ then take a unit vector $\bf n$ defining the direction of the segment into which you want to project, and take the dot product to arrive to $$ s(t) = a'\sin \left( {\omega t + \beta } \right) = a'\sin \left( {\omega \left( {t - t_0 } \right)} \right) $$ where $a'=a$ if $\bf n$ is in the $x,y$ plane.