PHP Ellipse Circumference Function - Numerical Solution to the Complete Elliptic Integral of the second kind

PHP Ellipse Circumference Function

This PHP function computes the circumference or perimeter of an ellipse, spheroid, circle, or sphere by evaluating the complete elliptic integral of the second kind using an infinite power series summation based on Legendre polynomials.

Computing the circumference of a circle is easy, but when it takes the form of an ellipse, there is no simple formula and the solution to an infinite series is required to do it. This is an extremely difficult problem to work out manually.

POSSIBLE APPLICATIONS:

Computing the length of an ellipse such as an elliptical planetary orbit or computing the equatorial and/or polar meridian lengths of any oblate spheroid like the planet Jupiter or Saturn, etc.

This function solves the problem using arbitrary-precision arithmetic.

The function only requires two parameters, the semi-major and semi-minor axes of the ellipse or spheroid, and returns the elliptical circumference to up to 16 decimals.

Author:
Jay Tanner - 2021

Development Language:
PHP v7.4.9

Current Web Site
www.NeoProgrammics.com

License:

This program source code and any related data and graphics are released to the public domain
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In the screencap above, equation #6 is the problem solved by the function. Here are two applications on my website in which the ellipse circumference function is applied.

http://neoprogrammics.com/!abc/math-science-tools/earth-gps-ellipsoid-3d-xyz-calculator/

http://neoprogrammics.com/Basic_Spheroid_Calculator.php

Below is a listing of the PHP function, also included in the archive with a demo program.
 

/*
   Ellipse Circumference Function

   Author : Jay Tanner - 2019
   License: Public Domain

*/

   function Ellipse_Circum ($RParam='1', $rParam='1')
{
   $R = trim($RParam);
   $r = trim($rParam);

   if (!is_numeric($R) or !is_numeric($r))
      {return 'ERROR: Invalid argument. Arguments must be purely numeric.';}

   $TwoPi = '6.28318530717958647692528676655900576839433879875';

   $decimals = 16;  $d = $decimals + 5;

   if (bccomp($R, $r, $d) < 0)  {$w=$R;  $R=$r;  $r=$w;}

   $e2 = bcsub('1', bcdiv(bcmul($r,$r, $d), bcmul($R,$R, $d),$d),$d);

   $n = $e2n = $PrecisionLevel = 1;   $sum = 0;

   $PrecisionLimit = '0.' . str_repeat('0', $d-1) . '1';

   while (abs($PrecisionLevel) > $PrecisionLimit)
  {
   $N = 2*$n-1;  $Y=1;  for($n=0;  $n <= ceil($N/2) - 1;  $n++)
  {$Y = bcmul($Y, $N - 2*$n);}  $Y2 = bcmul($Y,$Y,$d);

   $N = 2*$n;    $X=1;  for($n=0;  $n <= ceil($N/2) - 1;  $n++)
  {$X = bcmul($X, $N - 2*$n);} $X2 = bcmul($X,$X,$d);

   $e2n            = bcmul($e2n, $e2,          $d);
   $E              = bcmul($Y2,  $e2n,         $d);
   $F              = bcmul($X2,  2*$n-1,       $d);
   $CurrSumTerm    = bcdiv($E,   $F,           $d);
   $sum            = bcadd($sum, $CurrSumTerm, $d);
   $PrecisionLevel = bcsub($CurrSumTerm, $PrecisionLimit, $d);
   $n++;
  }
   $CurrSumTerm = bcdiv(bcmul($Y2,$e2n, $d), bcmul(2*$n-1,$X2, $d),$d);

   $w = bcmul($TwoPi, bcmul($R, bcsub('1', $sum, $d), $d),$d);

   return bcadd($w, '0.'.str_repeat('0', $decimals).'5', $decimals);

} // END OF  Ellipse_Circum (...)