Circumference of an Ellipse - Proposed Method

There is no simple equation for the circumference of an ellipse, though there have been many approximations. It should be noted that for any given ellipse with major axis a and minor axis b that the ratio of the circumference to the diameter is a constant value and ranges from 2.0 to π. This is similar to the ratio π which is just an ellipse where a equals b. Therefore, a new function should be adopted into mathematics representing this ratio for the ellipse based on the initial ratio of b/a – denoted as ρ (rho). This can be defined as ePi(ρ) representing the “Elliptical p” based on ρ, shown as follows:

ePi(ρ) - The value returned by this function would be from a lookup table (See Appendix H).

The benefit is that now you can calculate the value of the circumference for any ellipse as follows:

Introducing standard functions into mathematics is not unprecedented where values are looked up from a table, such as with trig or log functions. Regarding the lookup table presented in Appendix H, the values were derived by a program summing successive triangle sides along a quadrant of the ellipse then multiplied by 4 – See Figure 1 – Successive Approximations for the circumference of an Ellipse. The generated table was based upon 1000 elliptical configurations where ρ ranged from 0.0 to 1.0. Each entry in the table was generated using 100,000 triangles for each value of ρ utilizing 14-digit decimal resolution. The functionality behind ePi(ρ) that uses a lookup can perform a linear interpolation for values between a given value of ρ (rho).

Figure 1 – Successive Approximations for the circumference of an Ellipse