I'm writing a piece of software that will manipulate 2 servos to trace (hopefully) the outline of hypo/epicycloids using mapped pairs as coordinates where:
x = $ (R-r)\cos \ \theta +z \cos \theta \left(\dfrac{R-r}{r} \theta\right)$
y = $ (R-r)\sin \ \theta -z \sin \theta \left(\dfrac{R-r}{r} \theta\right)$
In the code, they look like this:
$pos_x=((R-r)*\cos(z)+d*\cos(((R-r)/r)*z));$
$pos_y=((R-r)*\sin(z)-d*\sin(((R-r)/r)*z));$
where
R = radius of the larger circle,
r = radius of smaller circle,
z = a fixed point the center of the smaller circle
I can fix this mechanically by only triggering the laser diode at each ordinate pair, and turning it off as it transitions to the next pair. However, I am hopeful there is a variation of the above equations that will do it more elegantly.
Failing that, what might be the best way to use the mapped coordinates to create only the contiguous line that is the outline of the figure?
Thanks!
Your equation is hypotrochoid:
$$ \begin{bmatrix} x \\ y \end{bmatrix}= \begin{bmatrix} (R-r)\cos \theta+d\cos \left(\frac{R-r}{r} \theta \right) \\ (R-r)\sin \theta-d\sin \left(\frac{R-r}{r} \theta \right) \end{bmatrix}$$
Equation of hypocycloid should read:
$$ \begin{bmatrix} x \\ y \end{bmatrix}= \begin{bmatrix} (R-r)\cos \theta+r\cos \left(\frac{R-r}{r} \theta \right) \\ (R-r)\sin \theta-r\sin \left(\frac{R-r}{r} \theta \right) \end{bmatrix}$$
You plotted the area confined by $(\theta,d)\in [0,2\pi] \times [0,r]$.
Replacing $d$ by $r$ and plotting $\theta \in [0,2\pi]$ will give a curve.