x(u,v), y(u,v), z(u,v) parametric equations for a special cycloid

Question

I'm trying to find out 3D parametric equations for a cycloid. I know that a cycloid is a 2D curve that is generated by a point on a rolling circle.

But my circle is rolling around another circle, and both of these circle axes are perpendicular to each other. The second circle is stationary and the first circle with the point is rolling around the fixed one. I need to get an equation for the curve generated by the point.

Please avoid vectors.

The 2D parametric equation for the cycloid is

$$\begin{align*}x(t)&=rt-r\sin(t)\\ y(t)&=r-r\cos(t)\end{align*}$$

where $r$ is the radius.

I need this

$x(u,v)=?\\ y(u,v)=?\\ z(u,v)=?$

I have been trying to get this for a year for my project. 3D equations are tough. Any help would be appreciated. thanks

Picture to show my problem

Answer 1

From the two-dimensional parametrization, we simply have $$\begin{align*} x(t) &= R \cos \bigl( \tfrac{r}{R}(t - \sin t) \bigr), \\ y(t) &= R \sin \bigl( \tfrac{r}{R}(t - \sin t) \bigr), \\ z(t) &= r - r \cos t. \end{align*}$$ Suitable choices for $r$ and $R$ will yield a closed curve, in which case the appropriate range for $t$ is $t \in [0, 2\pi R/r).$

Answer 2

This previous answer displays the explicit parametric equations for the spherical epitrochoid, which I will reproduce here for completeness:

$$\begin{align*}x&=(p+r\cos\;\omega)\cos\;t-h r\left(\cos\;\omega\cos\left(\frac{p}{r}t\right)\cos\;t-\sin\left(\frac{p}{r}t\right)\sin\;t\right)\\y&=(p+r \cos\;\omega)\sin\;t-h r\left(\cos\;\omega\cos\left(\frac{p}{r}t\right)\sin\;t+\sin\left(\frac{p}{r}t\right)\cos\;t\right)\\z&=(r\sin\;\omega)\left(1-h\cos\left(\frac{p}{r}t\right)\right)\end{align*}$$

where $p$ is the radius of the fixed circle ("pitch circle"), $r$ is the radius of the rolling circle, $\omega$ gives the tilt of the rolling circle (for the OP's particular case, $\omega=\frac{\pi}{2}$), and $h$ gives the fraction of the distance from the rolling circle's center (in the cycloidal case the OP is interested in, $h=1$).