equation for shape of inner lunar terminator?

The inner lunar terminator, the curve that separates the bright and dark
parts of the moon's disc as viewed from the earth, is semi-elliptical in
shape.

We can model the disc as a unit circle centred on the origin, in which the
inner terminator runs from (0,1) to (0,-1) and its major semi-axis always
has length 1.

Assuming orbits are circular, how does the minor semi-axis of this ellipse
vary during the lunar cycle?

Thanks!

Harry