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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 and then generalize it. Consider sphere of radius r. Increase the radius by a distance d. The new volume is 43π(r+d)3=43π(r3+3r2d+3rd2+d3)=43πr3+4πr2d+4πrd2+43πd3 . So the volume of the coating (shell) is VolumenewVolumeold=Volumecoat=4πr2d+4πrd2+43πd3 or in terms of the polynomial d, Volumecoat=(old surface area×d)+(πmean length×d2)+(43π×d3) , 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: =1πSH dA For a sphere, the mean curvature is H=1/2(1/r+1/r)=1/r. The mean width is then =1/(π)1/r4πr2=4r, which is twice the more intuitive average Euler width of 2r. This gives the second term on the right hand side of equation 3 as 4πrd2, which agrees with second term on the right-hand side of equation 2.

For an ellipsoid in standard form, (xa)2+(yb)2+(zc)2=1 The volume is V=43π abc From the first fundamental form for the ellipsoid, the surface area is A(a,b,c)=SEGF2=abc2π0π0(a2cos2v+b2sin2v)sin2u+c2cos2usinu dudv=2πc2+2πabsin(ϕ)[E(ϕ,k)sin2(ϕ)+F(ϕ,k)cos2(ϕ)] where cos(ϕ)=c/a, k2=a2/b2(b2c2)/(a2c2), F(ϕ,k) is the Legendre form of the first incomplete elliptic integral, and E(ϕ,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 (a,b,c)=1πSeG2fF+gEEGF2 dA=1π2π0π0(a2cos2v+b2sin2v)sin2u+c2cos2usinu dudv=abcπA(1a,1b,1c) . 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, =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.


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