The Earth Analemma and Orthonormal Coordinate Systems

Reuel

Hi.

I have been researching the Earth-Sun analemma and I found this document about deriving the Earth-Sun analemma via orthonormal coordinate systems.

Unfortunately I do not know very much about orthonormal coordinate systems and while I understand the first bit about elliptical angles, I get a little confused throughout the rest of the document. It really starts to lose me around equation (11). Where does that come from? What is the logic behind it?

I have already looked at the web site Analemma.com and understood it but this other document on orthonormal systems looks to be a lot more accurate and sophisticated and less estimated.

If anyone would be willing to help me understand this paper (more or less starting with the introduction of orthonormal coordinates) I would greatly appreciate it.