Cococubed.com Coating an ellipsoid

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Contact: F.X.Timmes
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Let's start from something familiar and then generalize it. Consider sphere of radius r. Increase the radius by a distance d. The new volume is $$\dfrac{4}{3} \pi (r + d)^3 = \dfrac{4}{3} \pi (r^3 + 3 r^2 d + 3 r d^2 + d^3) = \dfrac{4}{3} \pi r^3 + 4 \pi r^2 d + 4 \pi r d^2 + \dfrac{4}{3} \pi d^3 \ . \label{1} \tag{1}$$ So the volume of the coating (shell) is $${\rm Volume_{new}} - {\rm Volume_{old}} = {\rm Volume_{coat}} = 4 \pi r^2 d + 4 \pi r d^2 + \dfrac{4}{3} \pi d^3 \label{2} \tag{2}$$ or in terms of the polynomial $d$, $${\rm Volume_{coat}} = ({\rm old\ surface\ area} \times d) + (\pi \cdot {\rm mean \ length} \times d^2) + \left(\dfrac{4}{3} \pi \times d^3\right) \ , \label{3} \tag{3}$$ which is Steiner's formula for any convex shape expanded by a distance d along the surface normals in 3D. Note growth along surface normals is not the same as scaling the object to a bigger size - only for a sphere are the two equivalent. An amazing fact is Steiner's formula for the polynomial in d is valid for any expanding convex shape - spheres, ellipsoids, cubes, whatever. For small $d$, the first term dominates - the thin shell approximation. Blow anything up large enough along the surface normals and it looks like a sphere, the third term. These two limits are connected by the second term, the "mean width", which geometrically is a mean curvature (units of 1/length) times a surface area: $$\ell = \frac{1}{\pi} \int_S H \ {\rm d}A \label{4} \tag{4}$$ For a sphere, the mean curvature is $H = 1/2 \cdot (1/r + 1/r) = 1/r$. The mean width is then $\ell = 1/(\pi) \cdot 1/r \cdot 4 \pi r^2 = 4 r$, which is twice the more intuitive average Euler width of $2 r$. This gives the second term on the right hand side of equation $\ref{3}$ as $4\pi r d^2$, which agrees with second term on the right-hand side of equation $\ref{2}$.

For an ellipsoid in standard form, $$\left ( \dfrac{x}{a} \right )^2 + \left ( \dfrac{y}{b} \right )^2 + \left ( \dfrac{z}{c} \right )^2 = 1 \label{5} \tag{5}$$ The volume is $${\rm V = \dfrac{4}{3} \pi \ a b c } \label{6} \tag{6}$$ From the first fundamental form for the ellipsoid, the surface area is $$\begin{split} A(a,b,c) & = \int_S \sqrt{EG - F^2} \\ & = a b c \int_0^{2\pi} \int_0^{\pi} \sqrt{ (a^{-2} \cos^2v + b^{-2} \sin^2v) \sin^2u + c^{-2} \cos^2u} \sin u \ {\rm d}u {\rm d}v \\ & = 2 \pi c^2 + \dfrac{2 \pi a b}{\sin(\phi)} \cdot [ E(\phi,k) \sin^2(\phi) + F(\phi,k) \cos^2(\phi) ] \, \end{split} \label{7} \tag{7}$$ 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. Presumably one has the tools to numerically calculate these elliptic functions, hence the surface area, to near the precision of the chosen arithmetic. Note when $a=b=c$ that this expression reduces to the surface area of a sphere.

Using the second fundamental form for the mean curvature, the mean width is $$\begin{split} \ell(a,b,c) &= \frac{1}{\pi} \int_S \frac{eG - 2fF + gE}{EG - F^2} \ {\rm d}A \\ &= \frac{1}{ \pi} \int_0^{2\pi} \int_0^{\pi} \sqrt{ (a^{2} \cos^2v + b^{2} \sin^2v) \sin^2u + c^{2} \cos^2u} \sin u \ {\rm d}u {\rm d}v \\ &= \dfrac{a b c}{\pi} \cdot A \left ( \frac{1}{a},\frac{1}{b},\frac{1}{c} \right ) \ . \end{split} \label{8} \tag{8}$$ Wild! The mean width of an ellipsoid is akin to the volume of the ellipsoid times the surface area evaluated at the curvatures. Note when $a=b=c=r$ that this reduces to the mean length of a sphere, $\ell = 4r$.

The tool coating.f90.zip implements the above equations to calculate the volume of a coating, expanding along its normal, of a triaxial ellipsoid. The $a=b=c$ degenerate case of a sphere is included.

Please cite the relevant references if you publish a piece of work that use these codes, pieces of these codes, or modified versions of them. Offer co-authorship as appropriate.