# Calculate circle's offset given distorted ellipse, when projected onto a sphere

I have a circle moving on a spherical surface. If the camera angle is fixed, is there a way to calculate the original circle's offset from the centre given the distorted ellipse formed when the circle moves to the edge of the sphere(as seen from the camera)?

I've looked into Listing's plane and the Tissot directrix(because I'm working with eyeball rotation), but neither seems to provide a clear solution to this. I've also looked papers regarding eyeball roation but they seem to focus on the physiological aspects rather than the math of the movement.

In other words: In the picture above, given one of the outer nine images as well as the middle one, I want to calculate how much the circle(i.e. iris) has moved/rotated from its position in the middle picture. I assume it has to do with comparing the distorted ellipse with the circle, but I can't seem to find out how. Is there some formula for this?

## migrated from mathoverflow.netOct 28 '16 at 10:05

This question came from our site for professional mathematicians.

• Can you assume the camera is "infinitely far away", i.e., at a distance much larger than the radius of the eye? – Andrew D. Hwang Oct 28 '16 at 10:20
• Unfortunately, no. My camera is very close to the eye. – Caife Oct 31 '16 at 3:04