Ellipse perimeter approximation: New formula with absolute relative error less than one ppm.

Introduction

The ellipse, whose Cartesian equation is: (x/a) 2 + (y/b) 2 = 1 is named here as the standard ellipse. 'a' and 'b' (a ≥ 𝑏 ≥ 0) are the major and minor radii of the ellipse. Its perimeter P (a, b) is given by the formula 𝑃(𝑎, 𝑏) = ∫ √𝑎 2 𝑠𝑖𝑛 2 𝜃 + 𝑏 2 𝑐𝑜𝑠 2 𝜃 𝑑𝜃 2𝜋 0 where (a cos θ, b sin θ), 0 ≤ θ < 2 π, is a parametric point on the ellipse. Due to the symmetry of the ellipse w. r. t. its axes, it is enough to evaluate the Quarter Perimeter (abbreviated here as QPM), and, denoted by Q (a, b), given by the integral:

Obviously, P (a, b) = 4* Q (a, b), Q (a, 0) = a, and, Q (a, a) = πa /2. As the definite integral for Q (a, b) could not be evaluated by direct integration for values of b other than b = 0 and b = a, approximate value for Q (a, b) is obtained by Numerical Methods for given a and b. Simpson's 1/3-Rule Method is popular in this regard. The QPM values used here are obtained by this method, by dividing the interval of integration [0, π /2] into 500 equal sub-intervals, so that the length of each sub-interval is h = π/1000. Then, the Absolute Relative Error vis-vis the exact values is to the order of h 4 . Therefore, the Absolute Relative Error in Simpson's 1/3-values used here is less than 10 -10 . In an article published two years ago, the author introduced a new method to approximate the perimeter of the ellipse. It was shown there that the Quarter Perimeter Q (a, b) of the standard ellipse follows Lagrange's first order partial differential equation in a and b. Therefore, Q (a, b), as a solution of the partial differential equation, has to be a function of (𝑎 𝑝 + 𝑏 𝑝 ) 1 𝑝 , p ≠ 0 and/or √𝑎𝑏 , which are two independent particular solutions of the partial differential equation Koshy KI [1] . Further, it was shown that the empirical formula:

approximates the perimeter, where k ≈ 0.4825159, and, p = ln (2) / ln (π/2 -k/8) ≈ 1.680647, with maximum Absolute Relative Error less than 6.0*10 -5 , that is, 'less than six centimetres per kilometre', for all eccentricities.

The outcome of further research on the topic is presented in this article.

Notations

Conventional notations and terminologies related to the standard ellipse are used in this article. Aspect Ratio of the ellipse is the ratio (b/a). Obviously, the eccentricity 'e' is

Materials and Methods

In all forms, the Relative Error is calculated based on Q (a, b; Sm). It is well-known from Numerical Mathematics [2] , that the Absolute Relative Error of Q (a, b; Sm), vis-a-vis the actual quarter perimeter value Q (a, b), is of the order of h 4 or 10 (-10) Erwin Kreiszig 2010 [2] .

Modified Koshy's Formula # I

Q (a, b; K1): = (𝑎 𝑝 + 𝑏 𝑝 )

where k = 0.440143 and p = ln (2)/ ln (/2 -0.440143/2 3.2 ).

This formula approximates the Quarter Perimeter of the ellipse with Relative Error limits as follows: between -9.6647809*10 -7 and 9.6781748*10 -7 , for b/a ∈ [0.37, 1]; between 1.514497*10 -6 and 8.3793697*10 -6 , for b/a ∈ [0.31, 0.37) Between 0.31 and 0, the Relative Error increases and reaches to the order of 10 -4 . (Table 1)

In order to reduce the Relative Error for b/a < 0.

The maximum Absolute Relative Error in Q (a, b; Kc) is of the order of 10 -6 for all b/a from 0 to 1, and of the order of 10 - 7 for all b/a ≥ 0.29. (Table 2)

where k = 0.49496 and p = ln (2) /ln (/2 -0.49496/2 2.94 ). Equation (4) approximates the Quarter Perimeter of the ellipse with Relative Error lying in the interval (-4.9038243*10 -5 , 4.9045481*10 -5 ). Thus, the Absolute Relative Error is less than 5*10 -5 . Therefore, Q (a, b; K2) is a better formula than Q (a, b; Ko) for EPM Approximation. (Table 3.)

Discussion

Now that there are four formulae before us, academics may make their choice depending upon the precision preferred. It is pointed out here that formulae (1) to (4) are all functions of (𝑎 𝑝 + 𝑏 𝑝 )

1 𝑝 and √𝑎𝑏 and can be re-written in terms the Geometric Mean (GM:√𝑎𝑏) and the Arithmetic Mean (AM: (a+b)/2) of a and b. For, In (1), (ab) 2 / (a + b) 3 = (ab) 1.5 / (a + b) 3 * √𝑎𝑏 = (2GM/AM) 3 *√𝑎𝑏 ; in (2), (ab) 2.1 / (a + b) 3.2 = (ab) 1.6 / (a + b) 3.2 * √𝑎𝑏 = (2GM/AM) 3.2 *√𝑎𝑏 , and, in (4), (ab) 1.97 / (a + b) 2.94 = (ab) 1.47 / (a + b) 2.94 * √𝑎𝑏 = (2GM/AM) 2.94 *√𝑎𝑏.

References

1. Koshy KI. Ellipse Perimeter: A new Approximate Formula by a new Approach. International Journal of Engineering Science Invention Research and Development (IJESIRD). 2019;6(1);01-06