Imagine a solar eclipse here on Earth or some other planet (if you must). The moon perfectly covers the sun. From a 2D perspective a black circle covers the yellow circle, and both circles are the same radius.
Consider, at time = 0, the "sun circle" (I'm imagining a yellow piece of construction paper, for illustration) is completely visible. At t = 100 (lets say), the "moon circle" (I'm imagining a black piece of construction paper) is completely covering the "sun circle." There is some kind of curve to describe this.
It's similar to one square covering another square. If a yellow square is slowly covered (from one corner to the other) by a black square, from time t=0 to t=100, then at t=0 you can see the entire yellow square, at t=50 you can see 75% of the yellow square, and at t=100 you can see none of the yellow square. (Given by the formula (t)^2 / (t_f)^2; where t_f is "t final" or 100 in our example.) Likewise, the circle question would involve a curve who's second derivative was positive.
My question is, what model (or formula) describes the area visible on the circle being covered?