Area Swept- From Vertex

As can be seen from the previous proof, the area of the ellipse can be derived for the three-dimensional based definition, providing the circle being rotated is along the x axis in the z direction. This property can be used to calculate an area swept be a vector in the ellipse from the center point. Given an ellipse with a known major and minor axis, the area swept from the center point can be calculated for any angle θ – See Figure 1- Area Swept from Center Point.

Figure 1- Area Swept from Center Point

The first step in the process is to calculate the value of R (vector magnitude) based on the given angle. The equation used for this is shown in Appendix E.


The next step will be to transform the ellipse by rotating it from the z dimension on the x axis back to the same plane as the current ellipse – See Figure 5 - Ellipse Area Swept from Center Vertex Transformed. Once transformed back to the same plane as the ellipse, the value of Y will increase by the ratio a/b to be Y’. Note that the Y is transformed by the reciprocal of the b/a ratio as the rotation direction is reversed. The Y transformation also leads to a new angle on the circle denoted as θ’.

 

Calculating the value of Y and Y’ is shown in Equation 6 - Transforming the Y and the Angle, along with determining the transformed angle θ’.

Equation 6 - Transforming the Y and the Angle

Based on the equations of a circle, the Area (A’) can be calculated as a ratio of θ’ over 2p to the total area of the circle. Once A’ has been calculated, the final value of A for the ellipse can be calculated by simply applying the b/a ratio to A. Note that the Angle is in radians, and that ‘a’ is substituted for radius of the circle since they are equal

Area Calculation of A' Simplified Transformation With Substitution
       
Full Equation
 
 
Equation 7 ('Area Swept from Vertex')