Encouraged by Joseph O'Rourke ( and inspired by the discussion at Thales' semicircle theorem in higher dimensions ), I ask about level sets in three dimensional space occuring from considering how big an object looks from different points.

As mentioned in the post above, Thales shows that for a line segment to appear to subtend a given planar angle q, a two-dimensional eye has to be at some point on a circular arc in order to see the segment appear that large. Joseph was hoping for something similar in three or higher dimensions, involving a hemisphere as a possible locus for an observer to view a disk and have that disk occupy the same solid angle (appear to have the same area, if not the same shape), regardless of the viewer's position on the hemisphere.

So I ask, for two test regions, the same question: where must an observer place their eye to view a region R so that the solid angle subtended by R is a given value q? In Joseph's example, q was $(2 - \sqrt{2})\pi$ steradians, and R was the base of a hemisphere with the observer at its apex looking down. My two test regions are Joseph's disk, and a region which is two congruent rectangles separated by a small distance. You can substitute a block letter capital H instead, with an arbitrarily thin cross member. Also, pick your q greater than 0 and fix it before you start computing the locus.

For the disk, I imagine the locus will be a circular ellipsoid minus an equatorial circle. I currently reserve my guess on the locus for the other region.