PHP Function for Elliptical or Circular Circumference or Perimeter

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<?php

/*
 This program demonstrates a possible use of the ellipse circumference
 function defined below.  It can be applied to 2-dimensional ellipses
 or the cross-sections of 3-dimensional ellipsoids.

 According to the reference ellipsoid (WGS-84) used by the GPS system, the
 equatorial radius of the Earth is 6378.137 kilometers and the polar radius
 is 6356.752314245 kilometers.

 In this demo, the function is used to compute the vertical polar circumference
 or the perimeter of any general meridian such as the prime meridian.  It will
 work for any ellipsoid Earth or planetary model or any general ellipsoid as
 long as the eccentricity is not too extreme (e.g., Length/Width  ≤  4.5).

 For Earth (WGS-84):
 R = 6378.137 km = Equatorial radius
 r = 6356.752314245 km = Polar radius

 Using the above parameters, the circumferences should work out to:
 Ce = 40075.0166855784861531 = Circular equatorial circumference
 Cm = 40007.8629172503278127 = Elliptical polar meridional circumference

*/

// Compute and display both equatorial and meridional circumferences.

// Define equatorial and polar radii.
   $R  = '6378.137';
   $r  = '6356.752314245';

// Compute equatorial circumference (r = R).
   $Ce = Ellipse_Circum ($R, $R);

// Compute meridional or polar circumference (R >= r).
   $Cm = Ellipse_Circum ($R, $r);

print
"<pre>
For WGS-84 Earth spheroid model:

Equatorial:
Radius = $R km
Circumference = $Ce km

Polar or Meridional:
Radius = $r km
Circumference = $Cm km
</pre>";


/*
   ###########################################################################
   This function computes the perimeter of an ellipse with the given
   parameters (R, r).  It is a high-precision, brute-force numerical
   solution to the complete elliptic integral of the second kind.

   Internally, arbitrary-precision arithmetic is used to help preserve
   accuracy, with the final result truncated at 16 decimals.

   NOTE:
   Since this program uses BC arithmetic, the numeric values used
   should be numeric strings, like "3210.123465978" to represent
   EXACT values and avoid confusion with the default fixed double-
   precision numbers.

   Author : Jay Tanner - 2025
   ###########################################################################
*/

   function Ellipse_Circum ($RParam='1', $rParam='1')
{
// -----------------------------------------------
// Read input (R, r) ellipse parameter arguments.
// they are assumed to be purely numeric strings.

   $R = trim($RParam); // Corresponds to (a)
   $r = trim($rParam); // Corresponds to (b)

// -----------------------------------------------------
// Return FALSE (ERROR) if either (R, r) is non-numeric.

   if (!Is_Numeric($R) or !Is_Numeric($r))   {return FALSE;}

// -----------------------------------------------
// Define internal constant (2*pi) to 50 decimals.

   $TwoPi = '6.28318530717958647692528676655900576839433879875';

// -----------------------------------
// Set to 16 decimals final precision.

   $decimals = 16;

// ----------------------------------------------
// Use 8 extra decimals for internal computations
// to reduce accumulative rounding errors in the
// final result.

   $d = $decimals + 8;

// ---------------------------------------------
// If (r > R), then swap values so that (R > r).

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

// ------------------------------------------------------
// Compute the square of the elliptic eccentricity value.

   $e2 = bcSub('1', bcDiv(bcMul($r,$r,$d), bcMul($R,$R, $d),$d),$d);

// -------------------------------------------------
// Perform power series summation loop while the
// nth term is > the 16-decimal (PrecisionLimit).
// This limit could be changed, but 16 decimals
// should generally be sufficient.

   $n = $ePow2n = $PrecisionLevel = 1;   $sum = 0;

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

   while (abs($PrecisionLevel) > $PrecisionLimit)
  {
   $N = 2*$n-1;  $X=1;  for($n=0;  $n <= ceil($N/2) - 1;  $n++)
  {$X = bcMul($X, $N - 2*$n);}  $X2 = bcMul($X,$X, $d); // (2n-1)!!²

   $N = 2*$n;    $Y=1;  for($n=0;  $n <= ceil($N/2) - 1;  $n++)
  {$Y = bcMul($Y, $N - 2*$n);} $Y2 = bcMul($Y,$Y,$d); // (2n)!!²

   $ePow2n         = bcMul($ePow2n, $e2, $d); // e^(2n)
   $E              = bcMul($X2, $ePow2n, $d); // (2n-1)!!² * e^(2n)
   $F              = bcMul($Y2, 2*$n-1,  $d); // (2n)!!² * 2n-1
   $CurrSumTerm    = bcDiv($E, $F, $d);
   $sum            = bcAdd($sum, $CurrSumTerm, $d);
   $PrecisionLevel = bcSub($CurrSumTerm, $PrecisionLimit, $d);
   $n++;
  }
   $CurrSumTerm = bcDiv(bcMul($X2,$ePow2n, $d), bcMul(2*$n-1,$Y2, $d),$d);

// ----------------------------------------------------
// Finish computing the full elliptic circumference and
// round-off result to the given number of  decimals.

   $w = bcMul($TwoPi, bcMul($R, bcSub('1', $sum, $d), $d),$d);

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

} // END OF  Ellipse_Circum (...)

?>

Below is the PHP code for only the ellipse circumference function, without all the comments.

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/*
   Ellipse Circumference Function

   Author : Jay Tanner - 2025
*/

   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 + 8;

   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 (...)