Subsolar point modelling on an equirectangular map projection

[nAUTILUS]

Hello fellow cartographers!

I am working on an almost tidally locked world. Its day length is equal to one year but it has some axial tilt. This causes the subsolar point to oscillate on a north south line during each year. See the attached illustration. The planet is on an eccentric orbit, which leads to another type of oscillation of the subsolar point, this time east to west.
My question is:
How do I map this movement (overall it's an ellipse) of the subsolar point to an equirectangular map? And how can I map a terminator (line separating night and day) to an equirectangular map?

Any help would be greatly appreciated!

Tilit concept.png

[nAUTILUS]

I kept working on it and just solved it by adding 4 points on my equirectangular map, two at a distance from the poles of the axial tilt and two more perpendicular to them at a distance of the angle of libration. Then I connected them with a sine wave. No idea if that is the correct shape but the results look right. I can upload another illustration if anyone has a similar project and would like to see it.