Check out the newest version of the paper: See

http://vixra.org/abs/1701.0614
I submitted the paper to a journal. I'm certain that they will reject the paper even before it gets to peer review, but you never know until you try, right?

Imagine having a 3D sample space, where you march along that space, taking a sample at every step. The sample equation is given in the paper... see equations 1 and 2. This (eventually) gives you a 3D triangle mesh that represents the event horizon. I use Marching Cubes for this tessellation: the iso value (also known as the threshold) that I use is 1, and that's where the event horizon is.

It turns out that the event horizon shape for two merging unit mass black holes resembles the Cassini ovals. Cassini ovals are a spiric section (vertical slices of a torus), and are visualized in the first 10 figures.

The last figure likely requires a generalization of Cassini ovals, where the torus being sliced is lopsided like a ring cyclide (Dupin cyclide). Attached is a PNG file, a picture of a ring cyclide. I used the implicit equation found in the link below, with a threshold value of 0:

https://en.wikipedia.org/wiki/Dupin_cyclide#Elliptic_cyclides
What can I describe for you to help you understand?