Yep, conic sections are a *lot* easier in polar; in fact, they *all* have the same equation! I have an advantage in that I have my orbital mechanics text (Curtiss) sitting right here.

The equation you posted is one form of the 'orbit equation', aka Kepler's First Law. The e is the eccentricity; if it is 0, you have a circle; if it is 1, you have a parabola; if it is > 1, you have a hyperbola; if it is between 0 and 1, you have an ellipse. This means that if we approach the math from a polar perspective, we get solutions for all the conic sections for free (neglecting issues of what snaps, constraints, etc we want).

The 'perifocal' frame is a reference frame centered at the focus (F), where the other focus would be F' (read as "F prime"). The point on the apse line (defined in R^2 as the line defined by the two foci) closest to F is periapsis, and r_p is the position of that point relative to F; r_p is the angle (theta) reference, i.e. 0 degrees (radians), with positive direction being counter-clockwise. N.B. 'r' is the notation we use for 'position', so r(theta) is the position of the point on the conic section that a line from F to the ellipse with the angle of theta (relative to our angle reference). In polar coords, position would be a two-dimensional vector of <r, theta>, which can be converted to a two dimensional vector in <x, y> as required.

Any point or equation can be translated back and forth between coord systems; my approach would be to use whatever coord system was easiest for the task at hand. I will say that I would find constraint(s) on F to be very useful (the focus of my parabolic reflector must be 'here' relative to this feature).

For example, we could use a perifocal frame to do everything we need to with a conic section, and then convert our results back to cartesian coords if we needed to do something that is easier in cartesian coords.

Anyhow, like I said, I still have no knowledge of the FC codebase yet, I was just wondering about the approach being taken with the math here.