Im trying to solve the following problem :
Given the ellipsoid represented by the matrix $\widehat{A}$ and knowing the coordinates of each point A,B,C,D,E,F,G,H, calculate the volume of intersection (see the figure below).
The ellipsoid centered at $\mu$ is given by : $$(X-\mu)^T \widehat{A} (X-\mu)=1$$
The volume of n-dimensional ellipsoid is given by its general formula:
$$V_n = \frac{2}{n} \frac{\pi^{n/2}}{\Gamma(n/2)}(a_1 \> a_2 \>... \> a_n)$$
where $\Gamma$ is Eurler's gamma function, $a_i$ are the semi-axis of ellipsoid calculated using eigenvalues ($\lambda_i$) of $\widehat{A}$: $$a_i = \frac{1}{\sqrt{\lambda_i}}$$
My real probleme is n-dimensional (not exactly 3) that's why need to stick with general formulas. Any ideas how to calculate the immersed volume for the case n=3? I wish could have Archimedes' enlightenment.
