Basic Oblate Spheroid Calculator

Here is the summation equation solved by the function to numerically evaluate the
circumference (C) followed by the PHP source code upon which this program is based.

Circumference of Circle, Ellipse or Oblate Spheroid Meridian

$C = Ellipse_Circum ($r, $R);

Double-Click to Select ALL Code



/*
   Ellipse Circumference Function

   This function is a high-precision numerical solution to the complete
   elliptic integral of the second kind.   It returns the circumference
   of an oblate spheroid or ellipse with parameters (r, R).

   Where: r = Polar radius of spheroid/ellipse (Lesser value)
            = Semi-minor axis of ellipse
         and
          R = Equatorial radius of spheroid/ellipse (Greater value)
            = Semi-major axis of ellipse

   If (r == R), then C = Circumference at equator = Circle
   If (r <> R), then C = Circumference of polar meridian = Ellipse

   (r, R) can be in any linear units as long as the same units are used
   for both parameters.  The returned result will be expressed in the
   same units (m, ft, km, mi, etc.) as used for (r, R).  If (r == R),
   then the form is a perfect sphere/circle, otherwise the form is an
   oblate spheroid/ellipse.

   EXTERNAL DEPENDENCIES: NONE

   NOTE:
   The function invokes arbitrary precision arithmetic, so the arguments
   should be expressed as numeric strings, such as '12.345' or "1.2345"
   so as not to be confused with the normal 16-digit, double-precision
   values.

   Author : Jay Tanner - 2024
   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: Both arguments (r, R) must be purely numeric.';}

// Define 2pi to 64 decimals.
   $TwoPi = '6.2831853071795864769252867665590057683943387987502116419498891846';

   $Q = 32;  $q = 16;

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

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

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

   $PrecisionLimit = '0.' . str_repeat('0', $Q-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,$Y,$Q);

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

   $e2n            = bcMul($e2n, $e2,          $Q);
   $E              = bcMul($Y2,  $e2n,         $Q);
   $F              = bcMul($X2,  2*$n-1,       $Q);
   $CurrSumTerm    = bcDiv($E,   $F,           $Q);
   $sum            = bcAdd($sum, $CurrSumTerm, $Q);
   $PrecisionLevel = bcSub($CurrSumTerm, $PrecisionLimit, $Q);
   $n++;
  }
   $CurrSumTerm = bcDiv(bcMul($Y2,$e2n, $Q), bcMul(2*$n-1,$X2, $Q),$Q);

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

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

} // END OF  Ellipse_Circum (...)

Known (R, r) Parameters of Planets in Kilometers and Miles Computational Data Source: NASA The equatorial and meridional circumferences were computed from the given (R, r) parameters using the above function. R = Equatorial Radius r = Polar Radius R ≥ r Equat C = Equatorial Circumference Diff C = (Equat C) − (Polar C) Polar C = Polar Circumference __________________________________________________________ Earth | 6378.1 , 6356.8 km | Equat C = 40074.8 km = 24901.3 mi R r | Diff C = 1326.0 km = 824.0 mi 3963.2 , 3949.9 mi | Merid C = 40007.9 km = 24859.8 mi ____________________|_____________________________________ The Moon (Luna) | 1738.1 , 1736.0 km | Equat C = 10920.8 km = 6785.8 mi R r | Diff C = 6.6 km = 4.0 mi 1080.0 , 1078.7 mi | Merid C = 10914.2 km = 6781.8 mi ____________________|_____________________________________ Mars | 3396.2 , 3376.2 km | Equat C = 21339.0 km = 13259.4 mi R r | Diff C = 62.8 km = 39.0 mi 2110.3 , 2097.9 mi | Merid C = 21276.2 km = 13220.4 mi ____________________|_____________________________________ Jupiter | 71492 , 66854 km | Equat C = 449197 km = 279118 mi R r | Diff C = 14449 km = 8978 mi 44423 , 41541 mi | Merid C = 434749 km = 270140 mi ____________________|_____________________________________ Saturn | 60268 , 54364 km | Equat C = 378675 km = 235298 mi R r | Diff C = 18309 km = 11377 mi 37449 , 33780 mi | Merid C = 360366 km = 223921 mi ____________________|_____________________________________ Uranus | 25559 , 24973 km | Equat C = 160592 km = 99787 mi R r | Diff C = 1836 km = 1141 mi 15882 , 15518 mi | Merid C = 158756 km = 98647 mi ____________________|_____________________________________ Neptune | 24764 , 24341 km | Equat C = 155597 km = 96683 mi R r | Diff C = 1326 km = 824 mi 15388 , 15125 mi | Merid C = 154271 km = 95859 mi