Basic Orbital Ellipse Parameters

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

   Where:
   r = Semiminor axis
   R = Semimajor axis

   (r, R) can be in any convenient units, such as AUs, kilometers, miles, etc.

   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 - 2026
   ###########################################################################
*/

   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 (...)