Model IV

\(\require{color}\newcommand{\arule}[1]{{\color{#1}\Rule{3em}{1ex}{0ex}\;}}\)

We can improve this model, by extending it to the more general situation where the generating ellipse is not always in vertical position and may rotate in space. For this we specify three angles \(\phi\), \(\Omega\), \(\mu\), that establish the orientation of the generating curve in space. They measure the rotation of the curve around, respectively:

(Case 1: \(\phi\)) vector orthogonal to the plane of the initial ellipse;

(Case 2: \(\Omega\)) \(OZ\) axis;

(Case 3: \(\mu\)) major axis of ellipse elipse.

1º Case:

It suffices to replace \(s\rightarrow s+\phi\) in each sine and cosine functions of the ellipse equation \(E_{1}(\theta ,s)\).

2º Case:

(click the image to see the corresponding applet)

It suffices to replace \(\theta \rightarrow \theta +\Omega\) in each sine and cosine functions of the ellipse equation \(E_{1}(\theta ,s)\).

Then, the ellipse equations - centered at origin - become \[E_{2}(\theta ,s)=\left\{ \begin{array}{lcl} \left( x_{e}\right) _{2}(\theta ,s)=r_{e}(s)e^{\theta \cot (\alpha )}\cos (s+\phi )\cos (\theta +\Omega ) \\ \left( y_{e}\right) _{2}(\theta ,s)=r_{e}(s)e^{\theta \cot (\alpha )}\cos (s+\phi )\sin (\theta +\Omega ) \\ \left( z_{e}\right) _{2}(\theta ,s)=r_{e}(s)e^{\theta \cot (\alpha )}\sin (s+\phi ) \end{array} \right.\]

that is, the new shell equation becomes \[C_{2}(\theta ,s)=\left\{ \begin{array}{lcl} x_{2}(\theta ,s)=\left[ A\sin (\beta )\cos (\theta )+r_{e}(s)\cos (s+\phi )\cos (\theta +\Omega )\right] e^{\theta \cot (\alpha )} \\ y_{2}(\theta ,s)=\left[ A\sin (\beta )\sin (\theta )+r_{e}(s)\cos (s+\phi )\sin (\theta +\Omega )\right] e^{\theta \cot (\alpha )} \\ z_{2}(\theta ,s)=\left[ -A\cos (\beta )+r_{e}(s)\sin (s+\phi )\right] e^{\theta \cot (\alpha )} \end{array} \right.\]

3º Case:

\[\arule{orange}=\left( z_{e}\right) _{2}(\theta ,s)\Longrightarrow\begin{cases} \arule{magenta}= & \left( z_{e}\right) _{2}(\theta ,s)\cos(\mu)\\ \arule{red}= & \left( z_{e}\right) _{2}(\theta ,s)\sin(\mu) \end{cases}\]

Looking to the previous ellipse by profile and rotating it by angle \(\mu\)), we get \[z_{3}(\theta ,s)=z(\theta)+\left( z_{e}\right) _{2}(\theta ,s)\cos(\mu)=-Ae^{\theta \cot (\alpha )}\cos (\beta )+r_{e}(s)e^{\theta \cot (\alpha )}\sin (s+\phi )\cos (\mu )\]

\[\arule{red}=\left( z_{e}\right) _{2}(\theta ,s)\sin(\mu)\Longrightarrow\begin{cases} \arule{ForestGreen}= & \left( z_{e}\right) _{2}(\theta ,s)\sin(\mu)\sin(\theta+\Omega)\\ \arule{magenta}= & \left( z_{e}\right) _{2}(\theta ,s)\sin(\mu)\cos(\theta+\Omega) \end{cases}\]

Observing the two previous pictures (where the last one represents the initial ellipse and its result after rotation), we conclude that \[x_{3}(\theta ,s)=x_{2}(\theta ,s)-\left[ \left( z_{e}\right) _{2}(\theta ,s)\sin (\mu )\right] \sin (\theta +\Omega )\] \[y_{3}(\theta ,s)=y_{2}(\theta ,s)+\left[ \left( z_{e}\right) _{2}(\theta ,s)\sin (\mu )\right] \cos (\theta +\Omega ).\]

Summarizing, the shell is modelled by function \[C_{3}(\theta,s)=\begin{cases} x_{3}(\theta,s)=D[A\sin(\beta)\cos(\theta) & +r_{e}(s)\cos(s+\phi)\cos(\theta+\Omega)\\ & -r_{e}(s)\sin(s+\phi)\sin(\mu)\sin(\theta+\Omega)]e^{\theta\cot(\alpha)}\\ y_{3}(\theta,s)=[A\sin(\beta)\sin(\theta) & +r_{e}(s)\cos(s+\phi)\sin(\theta+\Omega)\\ & +r_{e}(s)\sin(s+\phi)\sin(\mu)\cos(\theta+\Omega)]e^{\theta\cot(\alpha)}\\ z_{3}(\theta,s)=[-A\cos(\beta)+r_{e}(s) & \sin(s+\phi)\cos(\mu)]e^{\theta\cot(\alpha)} \end{cases}\]

where \(D\) is a new parameter that we add to establish the direction of coiling (positive 1=dextral, negative -1=sinistral).

Remark: Functions \(C_{1}(\theta,s)\) and \(C_{2}(\theta,s)\) are already good mathematical models for shells, simpler but not so accurate as \(C_{3}(\theta,s)\). The choice of the model depends on the precision rate we want for our representation of shells.

To understand better the effect of parameters variation in the final shape of the shell, look to the following applets.

There are many shells in nature with nodules, bumps and spikes on their surface. Is it possible to improve this model in order to cover also those types of shells?

Image retrieved from [2]