I am interested in drawing a crescent moon in a vector drawing program. Our moon is a sphere illuminated by the sun in a certain direction, and viewed from Earth in a rotated direction. Our view of the spherical moon (which is basically an orthographic projection) is obviously just a circle, but the curved shape of the terminator line is not as simple. From my research, this crescent shape that is both illuminated and observed is called a spherical lune.
The outside shape is a circle. I am interested in figuring out what the geometry of the inner circle (the terminator line) would be for any angle of offset between the direction of illumination and direction of observation, ranging from a 0° offset (full moon) to a 90° offset (where the terminator is a straight line bisecting the circle) to a 180° offset (new moon). In other words, I am looking for an equation representing the 2D curve of an orthographic projection of a line of longitude at some angle connecting the poles of a sphere.
After figuring out how this terminator line's curve is shaped, I am also interested in how to represent it with a compound Bézier curve.
In simple terms, what is the Bézier curve shape of a spherical lune? (Rather than a regular lune, which isn't what the crescent moon actually looks like.)
