Using which mathematical equation did the mathematicians design the Hagia Sophia?

Anthemius was the Byzantine architect who replaced the old church of Hagia Sophia at Constantinople. He described the construction of an ellipse with a string fixed at the two foci and he described the focal properties of the parabola.

http://www-history.mcs.st-and.ac.uk/history/Mathematicians/Anthemius.html

I do not recall ever having run into a claim or scholarly evidence that Anthemius of Tralles and Isidore of Miletus used any specific well-known equation in the design of Hagia Sophia. Checking through my own library and as well as several online sites, I've found no statement that they did so.

However, Anthemius was a mathematician specializing in geometry, and Isidore had studied physics. Mathematics in general would have played an important role in the design, as it normally does in the work of architects. This site suggests some influence from the theories of Archimedes and from Heron of Alexandria on vaults and arch supports:


http://www.metmuseum.org/toah/hd/haso/hd_haso.htm

It may be that methods of constructing ellipses were used, but not as an EQUATION surely? I think the concept of equations started with Descartes in the 17th Century.