### Model V

In case we want to add nodules, bumps and spikes to the shell, it suffices to add some parameters and to replace the function that represents the ellipse (generating curve) \[r_{e}(s)=\frac{1}{\sqrt{\left(\frac{cos(s)}{a}\right)^{2}+\left(\frac{sin(s)}{b}\right)^{2}}},\,\,0\leq s\leq 2\pi ,\]

by \[r_{e}(s)+r_{nod}(s,\theta),0\leq s\leq2\pi,\theta\geq0,\]

where \[r_{nod}(s,\theta)=\begin{cases} 0,\mbox{se }w_{1}=0\vee w_{2}=0\vee N=0\\ Le^{-\left(\frac{2(s-p)}{w_{1}}\right)^{2}}e^{-\left(\frac{l(\theta)}{w_{2}}\right)^{2}},\mbox{otherwise} \end{cases}\]

and \[L(\theta)=\frac{2\pi}{N}\left(\frac{N\theta}{2\pi}-Round\left(\frac{N\theta}{2\pi}\right)\right)\]

for the new parameters:

\(p\): angle that measures the position of nodule in the generating curve;

\(w_{1}\): width of nodule along the generating curve;

\(w_{2}\): width of nodule along the helicoidal;

\(N\): number of nodules per whorl.

**Important note:**
this model contains the previous case of shells with no nodules; it suffices to make \(L=0\).
The model assumes that at a certain instant, \(\theta_{0}\),
there is only one node being constructed, that is, the generating curve at
\(\theta=\theta_{0}\)
has only "one protuberence" with respect to the initial ellipse.

The variation of these new parameters has significant consequences in the shape of the shell. To observe this, you may play with the following applets: