The Moon will be considered as a perfect sphere, enlightened by the Sun and seen by a human from the surface of Earth.
To simplify all the study, we will consider the Sun as a punctual source of light: thus the enlightened part of the moon is a spherical cap. (The impact of the Sun as a bowl is that borders of the spherical cap are fuzzy.) We will also ignore the fact that surfaces near the poles of the Moon are less enlightened by surfaces on the equator of the Moon.
We will not consider cases when the Moon, the Earth and the Sun are strictly aligned: in such cases the Moon is either invisible (lunar eclipse) or the Moon is a dark disk equivalent to the projection of its surface to the Earth.
The question is for all other "classical" cases. We want to characterize the shape more precisely than "a perfect disk" or parts of a perfect disk. We don't look for numeric values (unless they provide important information) but designation of the shape of the visible part of the Moon. Are there any ellipses? quadratic shapes? ...
The Moon is in-between the Sun and the Earth and thus its enlightened part is not visible to the human (full-day on Earth). Easy case: no shape.
The Moon is almost behind the Earth but not exactly in the shadow of the Earth. The full spherical cup is visible from the Earth, but what shape does the human see? (It is not a perfect disk, and it is a bit smaller than if the Moon was fully visible.)
Quarter of Moon
Earth-Moon-Sun form a 90° angle, so the human sees exactly one half of the enlightened part of the Moon. It appears as a perfect disk almost half-cut (smaller than half-disk, since the spherical cap does not contain poles).
The human sees part of the enlightened part of the Moon by an angle which in first approximation appears like a perfect disk cut by an... ellipse? How can we be more precise in the description of these two curves forming the perimeter of the visible area of the Moon?
(Note. Sorry if some mathematical terms employed are not pure English, since I'm French: please correct me.)