From my previous posts, we have come to a point where we can simulate the glitter pieces reflecting the light in a conic region as opposed to reflecting the light as a ray, and I think it is more realistic that the glitter is reflecting the light in a conic region. This means that when optimizing for the light and camera locations simultaneously, we actually can get different locations from we pre-determined to be the actual locations of the light and camera. Now, we want to take this knowledge and move back to the ellipse problem...
Before I could get back to looking at ellipses using our existing equations and assumptions, I wanted to first test a theory about the foci of concentric ellipses. I generated two ellipses such that a, b, c, d, and e were the same for both, but the value of f was different. Then, I chose some points on each of the ellipses and tried to use my method of solving for the ellipse to re-generate the ellipse, which worked as it had in the past.
I then went to pen & paper and actually used geometry to find the foci of the inner ellipse:
I found the two foci to be at about (-13, 7.5) and (11, -7.5). Now, using these foci, I calculated the surface normals for each of the points I had chosen on the two ellipses (so pretend the foci are a light and camera). In doing so, I actually found that the calculated surface normals for some of the points are far different from the surface normals I got using the tangent to the curve at each point:
The red lines indicate the tangent to the curve at the point, while the green vector indicates the surface normal of the point if the light and camera were located at the foci (indicated by the orange circles).
Similarly, I calculated and found the foci for the larger ellipse to be at (-15.5, 9) and (13.5, -9), and then calculated what the surface normals of all the points would be with these foci:
Again, the red lines indicate the tangents and the green lines indicate the calculated surface normals.
While talking to Abby this morning, she mentioned confocal ellipses, and it made me realize that it is possible that there is a difference between concentric and confocal ellipses. Namely, I think that confocal ellipses don't actually share the same values of a,b,c,d,e...maybe concentric ellipses share these coefficients with each other. And I think that is where we have been misunderstanding this problem all along. Now I just have to figure out what the right way to view the coefficients is...:)