Encouraged by Joseph O'Rourke ( and inspired by the discussion at
 http://mathoverflow.net/q/201281/6094 ), 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.