I need to draw an $8$ pointed star for an art piece. I would like all the legs to be congruent and approximately this general shape:
However, I would prefer if all the legs connected smoothly, or almost smoothly, to each other (none or barely visible cusps between the legs.
I would also like the ratio between the length of a leg and the radius of the central "circle" to be large, perhaps around $5:1$ (basically I want the legs to look long and skinny).
Does anybody know how to construct an equation for such a curve? It would also be ok if I had the equation for one leg or half a leg (less desirable) provided that when I draw all the $8$ legs there isn't a strong cusp between the legs.
The following may help. It is a general equation in the complex plane for epi- and hypocycloids (Ref: Zwikker, C. (1968). The Advanced Geometry of Plane Curves and Their Applications, Dover Press).
$$z_n=(n+1)e^{i\omega}+e^{i(n+1)\omega}$$
For positive $n$ you get the convex epicycloids, where for negative $n$ you get concave hypocycloids. Thus, you want $n=-8$. Unfortunately, the cusps are not as deep as you indicate in your sketch. However, you might be able to massage some parameters into the above equation to get the desired result. (Although I've had no luck so far.)
UPDATE: EUREKA! I've got it.
Let us normalize $z=z_{-8}/\max|z_{-8}|$. Then, set $r=|z|^p$, where $p>1$, and $\theta=\arg(z)$. Finally, create $Z_{-8}=re^{i\theta}$.
The figure below shows the original (left) and modified (right) hypocycloids (for $p=3$).