Given an ellipsoid and angle theta (illustrated below), what is the equation to find the length of the opposite side, shown in red. The hypotenuse connects at the ellipsoid center and perimeter.
For the ellipsoid parameters, you can use the WGS84 reference ellipsoid. Would appreciate a complete example so I can verify that I understand your description of the solution.
Thanks very much in advance
The ellipse with nonempty interior and centered at the origin is the image of a unit circle under an invertible linear transformation. Formally, if $A = \begin{pmatrix}a&b\\c&d\end{pmatrix}$ is the matrix that represents this transformation, then the set $$ E = \{ Ax ~|~ \lVert x\rVert^2 = 1\}, $$ is an ellipsoid.
Let $x=(\cos\theta,\sin\theta)$ be a vector on the unit circle. If the ellipse is not rotated, then the matrix is just $A=\begin{pmatrix}w&0\\0&h\end{pmatrix}$, where $w,h$ are the nonzero "width" and "height" of the ellipse respectively (or, the lengths of its principle axes). So $y = Ax = (w\cos\theta, h\sin\theta)$, which contains the lengths of right-angled sides (you're interested in $w\cos\theta$).