Parametric Spiral Ball Animation with GeoGebra 5

This is that neat thing I said I was working on in GeoGebra 5.  I wanted to come up with a mathematical model that described this metal mixing ball. After playing around with some formulas and testing out properties, I chose some constants for a, n, and d for the base image.  For each frame p, GeoGebra took a “screen capture” of the curve after all t units in the domain [-π, π] were mapped.  This is the formula I came up with:

 

$latex \begin{bmatrix}\bold{x}\\
\bold{y}\\
\bold{z}
\end{bmatrix}=\begin{bmatrix}a\cdot \sqrt{1-t^{2}}\cos(n(t-\frac{p\pi}{d}))\\
a\cdot \sqrt{1-t^{2}}\sin(n(t-\frac{p\pi}{d}))\\
a\cdot t
\end{bmatrix}$ such that -\pi\leq t\leq\pi and a,n,p,d\in\mathbb{Z^{\textrm{+}}}.

*The image below should be moving.  If it is not, click the image below to see the original file animated.*

Parametric Spiral Ball

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