Interface

#include <codecogs/geometry/volume/ellipsoidal_cap.h>

using namespace Geometry::Volume;

	double	ellipsoidal_cap (double x, double a, double b, double c)[inline] Computes the volume of an ellipsoidal cap.
	Real	cc_ellipsoidal_cap (Real x, Real a, Real b, Real c) This function is available as a Microsoft Excel add-in.

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Function Documentation

This module calculates the volume of an ellipsoidal cap.

The general equation for the ellipsoid is:

where is the semi-axis (width), is the semi-axis (depth) and is the semi-axis (height).

Consider a point on the height of the ellipsoid such that . The plane parallel to going through the point will intersect the ellipsoid and determine a subsection.

The volume of the resulting ellipsoidal cap is given by:

The situation is described below, where the filled cap is the volume we want to calculate.

Example:

#include <stdio.h>
#include <codecogs/geometry/volume/ellipsoidal_cap.h>
 
int main()
{
  // the x semi-axis
  double a = 10.5;
  
  // the y semi-axis
  double b = 5.2;
 
  // the z semi-axis
  double c = 3;
 
  // display the lengths of the semi-axes
  printf("a = %.1lf\nb = %.1lf\nc = %.1lf\n\n", a, b, c);
 
  // display the volume for different values of x
  for (double x = c; x >= -c; x -= 0.5)
    printf("x = %4.1lf    V = %.1lf \n",
    x, Geometry::Volume::ellipsoidal_cap(x, a, b, c));
 
  return 0;
}

Output

a = 10.5
b = 5.2
c = 3.0
 
x =  3.0    V = 0.0
x =  2.5    V = 13.5
x =  2.0    V = 50.8
x =  1.5    V = 107.2
x =  1.0    V = 177.9
x =  0.5    V = 258.1
x =  0.0    V = 343.1
x = -0.5    V = 428.0
x = -1.0    V = 508.2
x = -1.5    V = 578.9
x = -2.0    V = 635.3
x = -2.5    V = 672.6
x = -3.0    V = 686.1

Parameters:

x	the coordinate which determines the ellipsoidal cap
a	the x semi-axis (width)
b	the y semi-axis (depth)
c	the z semi-axis (height)

Returns:

the volume of the ellipsoidal cap

Note:

the value of must satisfy the inequality .

Authors:

Eduard Bentea (November 2006)

Source Code:

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