Rock Wall: The Sequel

What if. . .

the prior (circular) rock wall actually laid along the perimeter of an ellipse? But the fact was obscured by the near complete demolition of the former wall and all but scanty remains indicating where it used to be.

I have been attempting to solve the problem of how to go about reconstructing an elliptical rock garden wall (not a circular one, which was presented in an earlier article) using only a portion of the ellipse that is visible. Take a look at the diagram below of the rock wall as it is currently visible:

The only part of the rock wall visible to the gardener is the portion starting at point H and running through point G and slightly past point E. The coordinates corresponding to point H, G, and E are given as close estimates determined by the gardener when the problem was plotted out graphically on a Cartesian plane. To the casual observer, this is not a circular wall and it can be entertained that it is elliptical instead, although it wouldn't have to be. But, if less of the wall were remaining, its original construction might not be as obvious. Let's assume for sake of argument that the original wall was indeed elliptical.

The dilemma is how does one go about reconstructing this elliptical wall based solely on the arc that is visible; that is to say, arc HGE? The mechanical process for determining the center of a circular wall in my previous article was fairly elementary and straightforward. However, to achieve a similar feat on an elliptical wall one would naturally think it would require the gardener to determine the foci of the ellipse based on what is visible to him/her at the present time. But, must the gardener determine the foci or can another method be employed to accomplish this from an analytical standpoint?

My challenge to readers of this article is to determine a method whereby one can reconstruct the elliptical wall based solely on two points that lie alone the curvature of the ellipse and the endpoint E which lies at the Origin. How would you approach this problem?

Please post your comments and/or solution in the comment section below. Anyone who comes up with a viable solution to the problem will have their solution published on my blog with credit given to him/her.

The solution to this problem can be found by following this link.