Figure 1

Deriving this equation is based upon the fact that we already have an equation to determine the area swept from the center vertex, given any angle θ.- See Equation 7. To use this equation, we create a vector from the center to the x,y coordinates as shown in Figure 2. We then divide the area into two parts, the area from the center (Ac shown in Figure 2) and the area of the triangle (At as shown in Figure 3 below). Subtracting these two areas provides the area A = Ac - At.

Figure 2	Figure 3

To use equation 7 to determine the value of Ac, we need to first determine the center angle θ’ which is basically the arctan of y over x, shown here:

Computing the area of the triangle defined by the three points F2, Center, and (x,y). The basic formula for the area of a triangle is used, which is 1/2 the height (y) times the base (c). Note that the value of c is known for any ellipse - See Appendix D: The equation for the triangle portion then (At) is shown as follows:

Note that the value of R is derived based on angle θ - See Appendix D .

For any given ellipse, the area swept from focal point F1, based on some angle θ, is given as:

Basic
Expanded