Spherical_Properties(...) - PHP Science Labs

PHPSL Function: Spherical_Properties( )

Spherical_Properties ($KnownRCAV, $Decimals=16)

Dependencies: NONE

The basic sphere has four properties:

$R$ = Radius value
$C$ = Circumference value
$A$ = Surface area value
$V$ = Volume value

Given any single one of the four spherical $('R~C~A~V')$ properties, this function will compute the other three unknown related spherical property values to up to $100$ decimals precision.

Default = $16$ decimals, unless otherwise specified.

The returned string will contain all four numerical $('R~C~A~V')$ values, including the input argument, separated by spaces or an error message if something is wrong, such as an invalid argument.

The Relevant Spherical Equations:

Given the radius $(R)$:
$ \begin{align*} C ~=~ 2\pi R ~~~~~~~~~~~~~~~~~~~~~~~~ A ~=~ 4\pi R^2 ~~~~~~~~~~~~~~~~~~~~~~~~ V ~=~ \frac{4\pi R^3}{3} \end{align*} $

To obtain the radius $R$ from the $C$, $A$ or $V$ values:

$ \begin{align*} R ~=~ \frac{C}{2\pi} ~~~~~~~~~~~~~~~~~~~~~~~~ R ~=~ \sqrt{\frac{A}{4\pi}} ~~~~~~~~~~~~~~~~~~~~~~~~ R ~=~ \sqrt[\large3]{\frac{3V}{4\pi}} \end{align*} $

NOTES:

The constant $\large\pi$ is defined internally to $110$ decimals.
The $('R~C~A~V')$ symbols are NOT case sensitive.

Below is the PHP source code for the spherical properties function.

Double-Click to select ALL code


   function Spherical_Properties ($KnownRCAV, $Decimals=16)
{
   $KVStr = strtoupper(str_replace(' ', '', trim($KnownRCAV)));

// _____________________________________________
// Define local  constant pi to 110 decimals and
// related derived values. These constants could
// also be made global.

   $PI   = '3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148';
   $PIx2 = bcmul('2', $PI, 110);
   $PIx4 = bcmul('4', $PI, 110);

// __________________________________________
// Read decimals value from 0 to 100 maximum.

   $decimals = abs(intval(trim($Decimals)));
   if ($decimals > 100) {$decimals = 100;}
   $d = $decimals + 5 + floor($decimals / 4);

// _________________________________________
// Assume initial ERROR state by default.  A
// successful operation will set it to FALSE.

   $ERROR_FLAG_SET = TRUE;

// ____________________________________
// Handle the case of known radius (R).

   if (substr($KVStr,0, 2) == 'R=')
  {
   $R  = substr($KVStr,2, strlen($KVStr));
   $R2 = bcmul($R,$R, $d);
   $R3 = bcmul($R, bcmul($R,$R, $d), $d);
   $C  = bcmul($PIx2, $R, $d);
   $A  = bcmul($PIx4, $R2, $d);
   $V  = bcdiv(bcmul($PIx4, $R3, $d), '3', $d);
   $R  = bcadd($R, '0', $d);
   $ERROR_FLAG_SET = FALSE;
  }

// ___________________________________________
// Handle the case of known circumference (C).

   if (substr($KVStr,0,2) == 'C=')
  {
   $C  = substr($KVStr,2, strlen($KVStr));
   $R  = bcdiv($C, bcmul('2', $pi, $d), $d);
   $R2 = bcmul($R,$R, $d);
   $R3 = bcmul($R, bcmul($R,$R, $d), $d);
   $A  = bcmul($PIx4, $R2, $d);
   $V  = bcdiv(bcmul($PIx4, $R3, $d), '3', $d);
   $C  = bcadd($C, '0', $d);
   $ERROR_FLAG_SET = FALSE;
  }

// __________________________________________
// Handle the case of known surface area (A).

   if (substr($KVStr,0,2) == 'A=')
  {
   $A  = substr($KVStr,2, strlen($KVStr));
   $R  = bcsqrt(bcdiv($A, $PIx4, $d), $d);
   $R2 = bcmul($R,$R, $d);
   $R3 = bcmul($R, bcmul($R,$R, $d), $d);
   $C  = bcmul($PIx2, $R, $d);
   $V  = bcdiv(bcmul($PIx4, $R3, $d), '3', $d);
   $A  = bcadd($A, '0', $d);
   $R  = bcadd($R, '0', $d);
   $ERROR_FLAG_SET = FALSE;
  }

// ____________________________________
// Handle the case of known volume (V).

   if (substr($KVStr,0,2) == 'V=')
  {
   $V = substr($KVStr,2, strlen($KVStr));
   $x = bcdiv(bcmul('3', $V, $d), $PIx4, $d);
   $r = sprintf("%1.16f", pow($x, 1/3));

   for ($i=1;  $i <= 100;   $i++)
  {
   $R = bcdiv(bcadd(bcmul('2',$r,$d),bcdiv($x, bcpow($r,'2',$d),$d),$d),'3',$d);
        if (bccomp($r,$R,$d) == 0) {break;} else {$r = $R;}
  }
   $C  = bcmul($PIx2, $R, $d);
   $A  = bcmul($PIx4, bcmul($R,$R, $d), $d);
   $R  = bcadd($R, '0', $d);
   $V  = bcadd($V, '0', $d);
   $ERROR_FLAG_SET = FALSE;
  }

// _______________________
// Check error flag state.

   if ($ERROR_FLAG_SET)
      {return "ERROR: Invalid or missing R, C, A or V argument.";}

// _________________________________
// Round-off all values to the given
// number of output decimals.

   $w = '0.' . str_repeat('0', $decimals) . '5';
   $R = bcadd($R, $w, $decimals);
   $C = bcadd($C, $w, $decimals);
   $A = bcadd($A, $w, $decimals);
   $V = bcadd($V, $w, $decimals);

// Done.
   return "$R $C $A $V";

} // END OF  Spherical_Properties (...)

Double-Click to select ALL code

Some example call results are demonstrated below.

NOTE: Within the input argument string, the $('R~C~A~V')$ symbols are NOT case sensitive and any spaces within the argument string are ignored as if they do not exist.

The user supplies a single known spherical parameter argument to the function and it computes the other three related unknown values and returns a string consisting of all four values in $('R~C~A~V')$ order, separated by spaces as shown in the examples below.


   print Spherical_Properties ('r =1.75');  // Default = 16 decimals
// =--> '1.7500000000000000 10.9955742875642763 38.4845100064749671 22.4492975037770641'


// --------------------------
   print Spherical_Properties ('C= 125.3025', 10);
// =--> '19.9425122567 125.3025000000 4997.6932840722 33222.1865242790'


// --------------------------
   print Spherical_Properties ('a  =     13.032125', 12);
// =--> '1.018363175085 6.398564539071 13.032125000000 4.423812064371'


// --------------------------
   print Spherical_Properties ('V=7.5', 6);
// =--> '1.214295 7.629640 18.529268 7.500000'

References:

http://mathworld.wolfram.com/Sphere.html
https://en.wikipedia.org/wiki/Sphere

Jay Tanner - 2024