# Thales Style Level Sets

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.

• I like your ellipsoid conjecture for the disk, although perhaps it will only be "ellipsoid-like." Mar 30 '15 at 11:35
• @Joseph, I suspect for the disk it will be a surface of revolution based on a curve simply derived from the graph that Douglas Zare provided to your question. I conjecture ellipsoid just to give people something to refute, even though it is a natural first guess. For a rectangular region, one could imagine a parameterized surface which sliced in one direction resembles circular arcs of measure alpha and sliced in an orthogonal direction has measure beta; I expect reality will be more complicated. In order to draw out the subtleties, I proposed an H region instead. Mar 30 '15 at 19:33
• Except the original motivation, is it that important to restrict the question to plane 2-dimensional figures ? (Actually the only sets for which I can answer the question are balls !) Mar 31 '15 at 20:09
• At some point an anologous problem observing 3d objects in 3d space can be considered, but then the steradial projection may vary for other reasons, complicating the analysis. There are enough subtleties in this version that I think it important to consider the vision problem in this special form. Mar 31 '15 at 22:02
• I can write down an explicit formula in finite time for plane polygons, but it will be ugly... Apr 1 '15 at 6:47