I guess precisely, I am refering to the curve which must be drawn upon a canvas to give the observer with one eye closed, the proper image of a Hoola-Hoop (tm? gosh...I didn't realize that...) wrapped about the observer. The ellipse, hyperbola, and parabola are all possible.
John Conway pointed out that without the canvas, the curve is not a conic section. However, on this, I'm not sure exactly what is really going on. I agree that the image on the eyeball is not a conic section, but it's not clear to me that the actual perception is a conic section, because the brain may be structured to compensate for this. Can someone enlighten me?
I recall hearing about an experiment done with monkeys where monkeys were injected with radioactive glucose, had there eyes fixed upon a bullseye target, and then killed. When their brains were dissected, the radioactive glucose literally made a pattern on the folds of the brain: one, a bullseye pattern, the other, a grid indicating a polar coordinate transformation.
(I am not requesting that someone do a similar experiment to enlighten me! However, for me, things are complicated even more because of glasses,...or are glasses designed not to change matters?)
In any case, finding examples of absolutely true hyperbolas in nature is probably impossible. (For instance, Jupiter messes up orbits.)