# Numerical Integration Over Two Regions of an Ellipsoid

I would like to perform a numerical integration over the surface of an ellipsoid $D$. The domain must be split in two by a plane intersecting the ellipsoid (the intersection is arbitrary), so that we may integrate a function $f$ over the first domain $D_1$ and a different function $g$ over the second domain $D_2$. I do not know how to numerically define $D_1, D_2$ in such a way as to permit efficient numerical integration over them.

## Attempt at a Solution

We may assume that the ellipsoid is oriented such that its axes coincide with those of the coordinate system. Then the ellipsoid is defined as the set of solutions to $$F(x,y,z)\equiv \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} -1$$ where $a,b,c$ are known constants (the semi-principal axes lengths). We may then parameterize the surface using a 2D mesh $U\times V$ where $U\equiv\{u_0=0,u_1,...,u_n=2\pi\}$ and $V\equiv\{v_0=0,v_1,...,v_m=\pi\}$. This permits a parameterization $X \equiv \{\mathbf{x}_{ij}\}$ of $D$ where $$\mathbf{x}_{ij} \equiv (a\cos u_i \sin v_j,\ b \sin u_i \sin v_j,\ c \cos v_j).$$ Given an intersecting plane defined by the solutions to $$G(x,y,x)\equiv \alpha x+\beta y + \gamma z - 1$$ we may then assign each point in $\mathbf{x} \in X$ to $D_1$ if $G(\mathbf{x}) < 1$ and to $D_2$ if $G(\mathbf{x}) \geq 0$, where the direction of inequalities is arbitrary in this example.

## Problem with Solution

Consider $D_1=\{\mathbf{y}_{ij}\}$ as computed above. There is an ordering induced by the indexing of $U$ and $V$, but because of the sorting algorithm described above, two arbitrary points $\mathbf{y}_{i,j}$ and $\mathbf{y}_{i,j+1}$ that are adjacent in the enumeration of $D_1$ are not, in general, geometrically adjacent on the surface. This precludes the implementation of any numerical integration that relies on geometrically adjacent points to create patches.

## Possible Alternative

There exist explicit representations of integrals over ellipsoids given in terms of two parameters (see here, for example). The domain in this case is an ellipse, and I expect that $D_1$ will consist of a small number of transections of this ellipse. This will nonetheless present strange shapes, but I am aware of the existence of algorithms designed with such problems in mind.

Is there a better way to solve this problem?