Bounding ellipse constrained to horizontal/vertical axes

Question

Context: I'm trying to clip a topographic map into the minimum-size ellipse around a number of wind turbines, to minimize the size of the map. The program doing this map clipping can clip in ellipses, but only ellipses with axes aligned along the x and y axes.

I know the algorithm for the bounding ellipse problem (finding the smallest-area ellipse that encloses a set of points).

But how do I constrain this algorithm (or make a different algorithm) such that the resulting ellipse is required to have its major axis oriented either horizontally or vertically, whichever gives the smallest ellipse -- and never at an angle?

Of course, this constraint makes the resulting ellipse larger than it "needs" to be to enclose all the points, but that's the constraint nonetheless.

Answer 1

The algorithm described here (referenced in the link you provided) is about solving the following optimization problem:

minimize log(det(A))
s.t. (P_i - c)'*A*(P_i - c)<= 1

One can extend this system of inequalities with the following constraint (V is the ellipse rotation matrix, for detailed info refer the link above):

V == [[1, 0], [0, 1]] // horizontal ellipse

or

V == [[0, -1], [1, 0]] // vertical ellipse

Solving the optimization problem with either of these constraints and calculating the square of the resulting ellipses will give you the required result.