Shells 
Model III
To build the surface of the shell, we "replace in a continuous way" each point of the structural helicoidal curve by a closed curva, centered at that point, contained in a vertical plane. This curve will be similar at all points of the helicoidal. We call it the generating curve.
Note that we are "replacing" each point by a curve (with an infinity of points), which forces the introduction of a new parameter: the variable \(s\). While parameter \(\theta\) indicates a point of the helicoidal, with \(s\) we represent each point of the corresponding generating curve at \(\theta\).
In general, the generating curve of the shells has the shape of an ellipse. The parametric equation of an elipse centered at the origin, in polar coordinates, is \[r_{e}(s)=\frac{1}{\sqrt{\left(\frac{cos(s)}{a}\right)^{2}+\left(\frac{\sin(s)}{b}\right)^{2}}},\,\,0\leq s\leq 2\pi ,\]
where
- \(a\) is the horizontal half-axis of the ellipse, at \(\theta=0\);
- \(b\) is the vertical half-axis of the ellipse, at \(\theta=0\).
Note that the generating curve expandes when \(\theta\) increases - which allows the shell inhabitant to grow inside it. We assume that the expanding rate of the curve is equal to the one of the structural helicoidal, that is, \[r_{c}(\theta )=e^{\theta \cot (\alpha )}.\]
To see the variation of the generating curve along time, see the following app.
Then, the equation of the generating curve, in polar coordinates and assumed to be centered at the origin, is given by \[R_{e}(\theta ,s)=r_{e}(s).r_{c}(\theta )=r_{e}(s)e^{\theta \cot (\alpha )},\theta \geq 0,0\leq s\leq 2\pi ,\]
(click at the image to open the app)
\[\arule{magenta}=R_{e}(\theta,s)\Longrightarrow\begin{cases} \arule{red}=R_{e}(\theta,s)\sin(s)\\ \arule{blue}=R_{e}(\theta,s)\cos(s)\Longrightarrow & \begin{cases} \arule{ForestGreen}= & R_{e}(\theta,s)\cos(s)\sin(\theta)\\ \arule{orange}= & R_{e}(\theta,s)\cos(s)\cos(\theta) \end{cases} \end{cases}\]
which is equivalent, in cartesian coordinates, to
\[E_{1}(\theta ,s)=\left\{ \begin{array}{lcl} \left( x_{e}\right) _{1}(\theta ,s)=r_{e}(s)e^{\theta \cot (\alpha )}\cos (s)\cos (\theta ) \\ \left( y_{e}\right) _{1}(\theta ,s)=r_{e}(s)e^{\theta \cot (\alpha )}\cos (s)\sin (\theta ) \\ \left( z_{e}\right) _{1}(\theta ,s)=r_{e}(s)e^{\theta \cot (\alpha )}\sin (s) \end{array} \right.\]
Finally, to get the shell equations it suffices to translate the ellipses in the previous formula to the corresponding points at the helicoidal. We get
\[C_{1}(\theta ,s)=H(\theta )+E_{1}(\theta ,s)=\\ =\left\{ \begin{array}{lcl} x_{1}(\theta ,s)=Ae^{\theta \cot (\alpha )}\sin (\beta )\cos (\theta )+r_{e}(s)e^{\theta \cot (\alpha )}\cos (s)\cos (\theta ) \\ y_{1}(\theta ,s)=Ae^{\theta \cot (\alpha )}\sin (\beta )\sin (\theta )+r_{e}(s)e^{\theta \cot (\alpha )}\cos (s)\sin (\theta ) \\ z_{1}(\theta ,s)=-Ae^{\theta \cot (\alpha )}\cos (\beta )+r_{e}(s)e^{\theta \cot (\alpha )}\sin (s) \end{array} \right.\]
