Surface Area of an Ellipsoid $AX^2 + BY^2+ CZ^2 + DXY + EXZ + FYZ - R_0^2= 0$

I am attempting to determine the surface area of an ellipsoid described by the equation:

$$AX^2 + BY^2 + CZ^2 + DXY + EXZ + FYZ - R_{0}^2 = 0 \; .$$

When it comes to an ellipse aligned with its axes at the origin of a Cartesian system, as given by $$\frac{x^2}{a^2} + \frac{y^2}{b^2}=1$$, the process is straightforward, and we swiftly find that $$A = \pi ab$$, where $$a$$ and $$b$$ represent the semi-major and semi-minor axes respectively.

However, for an ellipse not meeting these conditions, like the example mentioned here, the task becomes more challenging but remains achievable through algebraic manipulation.

I attempted to apply a similar approach to the equation $$AX^2 + BY^2 + CZ^2 + DXY + EXZ + FYZ - R_{0}^2= 0$$ representing an ellipsoid, but I have not been able to solve it. Can someone provide assistance with this question, please?

Edit:

I do not have any other information such as eccentricity, length of the axes, etc... I only have the equation above.

• In wikipedia there is article called Ellipsoid, where the formula of ellipsoid surface area expressed through axes lengths is given Commented Oct 19, 2023 at 10:36
• I don't have any additional information about the lengths of the axes. I just have that equation. I've edited my question. Commented Oct 19, 2023 at 10:42
• Are asking how to transform to axis an aligned form, or how to evaluate the surface of an axis-aligned ellipse. Also note that the 2D equivalent to the surface area is the perimeter which is not known analytically for an ellipse. So the surface of an ellipsoid is also not going to be known analytically. Commented Oct 19, 2023 at 14:43

• First find $$\lambda_1, \lambda_2, \lambda_3$$, the eigenvalues of

$$\begin{pmatrix} A & \frac{D}{2} & \frac{E}{2} \\ \frac{D}{2} & B & \frac{F}{2} \\ \frac{E}{2} & \frac{F}{2} & C \end{pmatrix}$$

• For triaxial ellipsoid,

$$0<\lambda_1<\lambda_2<\lambda_3$$

• Semi principal axes are

\begin{align} a &= \frac{R_0}{\sqrt{\lambda_1}} \\ b &= \frac{R_0}{\sqrt{\lambda_2}} \\ c &= \frac{R_0}{\sqrt{\lambda_3}} \\ \end{align}

• Surface area $$S$$ for $$a>b>c>0$$

\begin{align} S &= 2\pi \left[ c^{2}+\frac{bc^{2}}{\sqrt{a^{2}-c^{2}}}\, F\left( \cos^{-1} \frac{c}{a}, k \right)+b\sqrt{a^{2}-c^{2}}\, E\left( \cos^{-1} \frac{c}{a}, k \right) \right] \\ k &=\frac{a}{b}\sqrt{\frac{b^{2}-c^{2}}{a^{2}-c^{2}}} \end{align}

• See also the journal article by Leo R. M. Maas here.

• Thank you very much for you answer. What does F(cos−1ca,k) and E(cos−1ca,k) mean? What happens if the eigenvalues are negative? Commented Oct 19, 2023 at 13:13
• @RFeynman If some eigenvalues are negative it is not an ellipsoid. $F$ and $E$ can be found here.
– user
Commented Oct 19, 2023 at 14:19
• @RFeynman They are incomplete elliptic integrals of the first and second kind. Commented Oct 19, 2023 at 14:28
• Thank you very much! Commented Oct 19, 2023 at 14:48

There is a paper written in 1979 that discusses methods for approximating the surface area of a scalene (triaxial) ellipsoid.

• Here's the paper: ams.org/journals/mcom/1979-33-145/S0025-5718-1979-0514826-4/…
– Nate
Commented Oct 19, 2023 at 10:36
• I don't have information regarding the eccentricity or the lengths of the axes. I've edited my question, as I had forgotten to mention these details. Commented Oct 19, 2023 at 10:42
• It is interesting that you give these closed form formulas for this particular cases, but you should warn the asker that we are here in the general case where the ellipsoids are neither prolate nor oblate ... Btw : in the case of prolate or oblate ellipsoids, could the formulas you give be explained using the ¨Pappus-Guldin theorem ? Commented Oct 20, 2023 at 5:19