Shells 
Model II
Let us look at the shell again, but now as an object in \(\mathbb{R}^{3}\) (tridimensional object).
We want to find a curve in space that "seen from above" looks similar to an equiangular spiral but moves down continuously with respect to the horizontal plane that contains the initial point of the curve. A curve with such a property is an helicoidal.
To take advantage of the curve that we have already in the planar case, we suppose that the "transition" of the spiral to the helicoidal \(H(\theta )\) does not change the distance between the points of the curve and the origin of the referential, that is, \[r(\theta)=Ae^{\theta \cot (\alpha )},\theta\geq0,\] still represents the distance of each curve point to the origin.
(click at the image to open the app)
\[\arule{red}=r(\theta)\Longrightarrow\begin{cases} \arule{blue}= & r(\theta)\cos(\beta)\\ \arule{orange}= & r(\theta)\sin(\beta) \end{cases}\]
Then the height of each point in this curve is given by
\[z(\theta )=-r(\theta )\cos (\beta ),\,\,\theta \geq 0,\]
where \(\beta\) \( (0% %TCIMACRO{\UNICODE[m]{0xba}}% %BeginExpansion {{}^o}% %EndExpansion \leq \beta \leq 90% %TCIMACRO{\UNICODE[m]{0xba}}% %BeginExpansion {{}^o})% %EndExpansion \) is the enlargement angle of the helicoidal.
On the other hand, note that to get the equations \(x(\theta)\) and \(y(\theta)\) of the helicoidal curve it suffices to replace \(r(\theta)\) by \[r_{p}(\theta )=r(\theta )\sin (\beta ) \]
in the equations of the spiral.
To see what happens to the helicoidal when the parameters \(\alpha\), \(\beta\) and\(A\) are changed, see the following app.
In summary, the equation of the helicoidal curve \(H(\theta)=\left(x(\theta),y(\theta)\right)\) in cartesian coordinates is given by \[\left\{ \begin{array}{l} x(\theta )=r_{p}(\theta )\cos (\theta )=Ae^{\theta \cot (\alpha )}\sin (\beta )\cos (\theta ) \\ y(\theta )=r_{p}(\theta )\sin (\theta )=Ae^{\theta \cot (\alpha )}\sin (\beta )\sin (\theta ) \\ z(\theta )=-Ae^{\theta \cot (\alpha )}\cos (\beta ) \end{array} \right. .\]
We have already the mathematical model for the substructure of the shell. The model for the shell surface (that is, the walls of the shell) is still missing...
