What is the equation which describes the inner circle of the crescent that a celestial body displays when view at an angle from its light source, as a function of the crescent-cycle period? For instance, the cycle of Earth's moon?
I know that on a new moon the inner curve is coincident with the circumference, so the inner curve's radius is the moon's radius (let's call that 1 unit) and it's displacement from the moon's center is 0 units. After one week (one fourth of the cycle period) the inner curve's radius is infinite, and it's center is infinite distance from the center of the moon. The in-between values are not so clear, though!
Does r=1/cos(2x·π) describe the inner curve's radius (x would be the period)? It at least fits the end-values. How about determining the distance of the inner curve's center from the center of the moon?
I figure that this is a physics question and not a mathematics question as I am looking to describe a phenomenon that occurs in nature, which is only one of the infinite curves which could be described. If this question is better placed in mathematics.SE or astronomy.SE then it can be moved.