Ellipsoid

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An ellipsoid is a closed curved surface whose three main orthogonal sections are elliptical, that is, they are originated by planes containing two Cartesian axes in each plane.

In mathematics, it is a quadric analogous to an ellipse, but in three dimensions.

An ellipsoid is obtained by "deforming" a sphere, through a homological transformation, in the direction of its three orthogonal diameters.

By rotating an ellipse around one of its two axes, an ellipsoid of revolution or spheroid is obtained.

Cartesian equation of an ellipsoid

The equation of an ellipsoid with center at the origin of coordinates and axes coincident with the Cartesians, is:

{x^2 over a^2}+{y^2 over b^2}+{z^2 over c^2}=1

where a, b and c are the lengths of the semi-axes of the ellipsoid with respect to the x, y, z axes, respectively; are positive real numbers and determine the shape of the ellipsoid. If two of these semiaxes are equal, the ellipsoid is a spheroid; if all three are equal, it is a sphere.

Surface

The surface area of an ellipsoid is given by the following formula:

S = 2pileft(c^2+bsqrt{a^2-c^2}E(alpha,m)+frac{bc^2}{sqrt{a^2-c^2}}F(alpha,m)right),,!

where

{displaystyle alpha ={begin{cases}arccos left({frac {c}{a}}right);{textrm {achatado}};o;{textrm {escaleno}}\arccos left({frac {a}{c}}right);{textrm {alargado}}end{cases}},!}

is its angular eccentricity, $m=frac{b^2-c^2}{b^2sin^2(alpha)},!$ and $F(alpha,m),!$ , $E(alpha,m),!$ are the elliptical integrals of first and second species.

An approximate equation of its surface is:

S;approx 4pi!left(frac{ a^p b^p+a^p c^p+b^p c^p }{3}right)^{1/p},!

where p ≈ 1.6075. With this expression, a maximum error of ±1.061% is obtained, depending on the values of a, b and c. The value p = 8/5 = 1.6 is optimal for quasi-spherical ellipsoids, with a maximum relative error of 1.178%.

Volume

The volume of an ellipsoid is given by the equation:

$V=;frac{4pi}{3} abc,!$

Using Differential Geometry we can prove the above expression. It is known that the volume of a closed region Ω corresponds to the triple integral of the function f(x,y,z) = 1 and that if any change of coordinates is made (for example spherical) it must be multiplied by the Jacobian of Change of Variable and adapt the limits of integration.

$V_{Omega } = iiint_{Omega } dV = iiint_{Omega } left | JPsi (rho,thetavarphi) right |drho dtheta dvarphi!$

In this case, the change of variable is of the pseudospherical type, much more general than that of the sphere (for a logical reason, an ellipsoid with all its parameters a,b,c equal generates a sphere, that is, that the sphere is a particular ellipsoid with a high degree of symmetry). The limits of integration have also been defined.

Psi (rho,thetavarphi) = begin{cases} & x= a rho sinthetacosvarphi {: rho in left [ 0,1 right ] } \ & y= b rho sinthetasinvarphi {: theta in left [ 0,pi right ] } \ & z=c rho costheta {: varphi in left [ 0,2pi right ] } end{cases}!

To calculate the Jacobian the matrix in partial derivatives should be calculated with respect to $rho, theta, phi$ and the determinant of this square matrix three by three results:

$left | JPsi (rho,thetavarphi) right | = abccdot rho^2 sintheta!$

Therefore, the integral that must be solved, taking into account the above, is:

$abc int_{0}^{2pi } int_{0}^{pi } int_{0}^{1} rho^2 sintheta drho dtheta dvarphi,!$

Operand:

abc int_{0}^{1} rho^2drho int_{0}^{pi }sintheta dtheta int_{0}^{2pi }dvarphi = abccdot left [ frac{r^3 }{3}right ]_{0}^{1}cdot left [ -costheta right ]_{0}^{pi }cdot left [ varphi right ]_{0}^{2pi } =abccdot frac{1}{3}cdot 2 cdot 2pi = frac{4}{3}pi abc !

(Q,E,D)

An alternate demonstration can be made with Riemann sums. This consists of adding along the X-axis the areas of the cross-sections. As the cross section of an ellipsoid is an ellipse, its area is given by $A_{x} = pi z(x) y(x)!$ so the volume of the ellipsoid would be given by:

2 pi int_{0}^{a} z(x) y(x) dx

Again, since the cross sections are ellipses, we have:

z = c sqrt{1-left (frac{x}{a} right)^{2}}

y = b sqrt{1-left (frac{x}{a} right)^{2}}

Replacing:

2 pi int_{0}^{a} b c left (1-left (frac{x}{a} right)^{2} right) dx = frac{4}{3} pi a b c!

Another way to calculate the volume by adding the areas of the cross section along the X-axis, without resorting to the ellipse area formula, is to express the area of those sections as the integral ${displaystyle int zcdot dy}$ between the limits of the ellipse ${displaystyle {frac {x^{2}}{a^{2}}}+{frac {y^{2}}{b^{2}}} = 1}$ .

Then the double integral is obtained:

{displaystyle V=2int _{-a}^{a}[int _{-{frac {b}{a}}{sqrt {a^{2}-x^{2}}}}^{{frac {b}{a}}{sqrt {a^{2}-x^{2}}}}ccdot {sqrt {1-{frac {x^{2}}{a^{2}}}-{frac {y^{2}}{b^{2}}}}} dy ] dx}

This double integral can be simplified to

{displaystyle 8cint _{0}^{a}[int _{0}^{{frac {b}{a}}{sqrt {a^{2}-x^{2}}}}{sqrt {1-{frac {x^{2}}{a^{2}}}-{frac {y^{2}}{b^{2}}}}} dy ] dx}

The inner integral (between square brackets), which represents the area of each cross section ${displaystyle int _{0}^{{frac {b}{a}}{sqrt {a^{2}-x^{2}}}}{sqrt {1-{frac {x^{2}}{a^{2}}}-{frac {y^{2}}{b^{2}}}}} dy}$ is solved without great difficulty ${displaystyle {frac {bpi }{4a^{2}}}cdot (a^{2}-x^{2})}$

Cuya integral entre ${displaystyle [0,a]}$ equals ${displaystyle {frac {bpi }{4a^{2}}}cdot {frac {2}{3}}a^{3}}$ ; and including and grouping all terms you get the volume formula: