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Coating an ellipsoid

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Contact: F.X.Timmes
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Let's start from something familiar. Consider sphere of radius r. Increase the radius by a distance d. The new volume is \begin{equation} \dfrac{4}{3} \pi (r + d)^3 = \dfrac{4}{3} \pi (r^3 + 3 r^2 d + 3 r d^2 + d^3) = \dfrac{4}{3} \pi r^3 + 4 \pi r^2 d + 4 \pi r d^2 + \dfrac{4}{3} \pi d^3 \ . \end{equation} So the volume of the coating is \begin{equation} {\rm Volume_{new}} - {\rm Volume_{old}} = {\rm Volume_{coat}} = 4 \pi r^2 d + 4 \pi r d^2 + \dfrac{4}{3} \pi d^3 \label{1} \tag{1} \end{equation} or \begin{equation} {\rm Volume_{coat} = Old\ Surface\ Area \times d + \pi \cdot mean \ length \times d^2 + \dfrac{4}{3} \pi \times d^3 } \ , \label{2} \tag{2} \end{equation} which is Steiner's formula for any convex shape expanding along the surface normals in 3D. Note growth along surface normals is not the same as simply scaling the object to a bigger size - only for a sphere are the two equivalent. An amazing thing about Steiner's formula is that the polynomial in d is valid for any expanding convex shape - spheres, ellipsoids, cubes, whatever.

For small d, the first term dominates, old surface_area $\times$ d. Blow anything up large enough and it looks like a sphere, the third term. These two limits are connected by the second term, the "mean length", which geometrically is essentially a volume to surface area ratio. \begin{equation} {\rm Mean \ length = \dfrac{3}{4 \pi^2} \cdot Old \ Volume \cdot Old \ Surface \ Area \ evaluated \ at \ the \ radius \ of \ curvature. } \label{3} \tag{3} \end{equation} For a sphere, the radius of curvature is $1/r$, and the mean length is $ 3/(4\pi^2) \cdot 4/3 \pi r^3 \cdot 4 \pi / r^2 = 4r$, which gives the second term in equation $\ref{2}$ as $4\pi r d^2$, which agrees with second term in equation $\ref{1}$. For an ellipsoid in standard form, \begin{equation} \left ( \dfrac{x}{a} \right )^2 + \left ( \dfrac{y}{b} \right )^2 + \left ( \dfrac{z}{c} \right )^2 = 1 \label{4} \tag{4} \end{equation} The volume is \begin{equation} {\rm V = \dfrac{4}{3} \pi \ a b c } \label{5} \tag{5} \end{equation} The surface area is \begin{equation} {\rm SA(a,b,c) = 2 \pi c^2 + \dfrac{2 \pi a b}{sin(\phi)} \cdot [ E(\phi,k) \sin^2(\phi) + F(\phi,k) \cos^2(\phi) ] } \label{6} \tag{6} \end{equation} where $\cos(\phi) = c/a$, $k^2 = a^2/b^2 \cdot (b^2 - c^2) / (a^2 - c^2)$, $F(\phi,k)$ is the Legendre form of the first incomplete elliptic integral, and $E(\phi,k)$ is the legendre form of the second incomplete elliptic integral.
The all-important mean length, per equation $\ref{3}$, is \begin{equation} {\rm Mean \ Length = \dfrac{a b c}{\pi} \cdot SA \left ( \frac{1}{a},\frac{1}{b},\frac{1}{c} \right ) } \label{7} \tag{7} \end{equation} which redues to the spherical case of $4r$ for $a=b=c=r$.

This code implements the above equations to calculate the volume of a coating around a general triaxial ellipsoid, including the $a=b=c$ degenerate case of a sphere.


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