Hello, Ferguzz!

Imagine a rotating ellipse.

A bounding box is placed around the ellipse oriented always parallel to the axes.

How can I calculate the width, height and angle of rotation of this ellipse?

The information I do have is the bounding box dimensions and the area of the ellipse.

I don't understand the difficulty.

We have an ellipsoid of revolution.

The box has length $\displaystyle L$ and height $\displaystyle H.$

The width of the box equals the height: $\displaystyle W = H.$

Code:

: - - - - - L - - - - - :
= *---------*-*-*---------* =
: | * : * | :
: | * : * | :
: |* : *| b
: | : | :
: * : * :
H * - - - - - * - - - - - * =
: * : * :
: | : | :
: |* : *| b
: | * : * | :
: | * : * | :
= *---------*-*-*---------* =
: - - a - - | - - a - - :

The "width" of the ellipsoid is $\displaystyle L.$

The "height" of the ellipsoid is $\displaystyle H.$

An equation of the ellipse is: .$\displaystyle \frac{x^2}{a^2} + \frac{y^2}{b^2} \:=\:1$

. . where: .$\displaystyle a = \tfrac{L}{2},\;b = \tfrac{H}{2}$

"The angle of rotation" has no meaning.