I would like to find the average among a number of ellipses.
The ellipses all have the same center, and the same major axis length, but they have different eccentricities and different orientations, which is what makes the problem challenging.
Averaging two ellipses with the same eccentricity, but with an angle of $pi/2$ between the major axes should result in a circle, as should the average between three same ellipses oriented at $pi/3$ from one another. Averaging two ellipses with the same eccentricity, but with an angle of $pi/4$ between the major axes should result an ellipse with a smaller major axis right in between the two axes.
Initially, it seemed that drawing a new ellipse through the intersection points between two ellipses would work, but that is not easily adapted to more than two ellipses, and it doesn't appear to work well if the eccentricity is different between ellipses.