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MathsGeometryVolume

ellipsoidal cap

Computes the volume of an ellipsoidal cap.
Controller: CodeCogs

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Interface

C++
HTML

Ellipsoidal Cap

 
doubleellipsoidal_capdoublex
doublea
doubleb
doublec )[inline]
This module calculates the volume of an ellipsoidal cap.

The general equation for the ellipsoid is:

where a is the width in the x-axis, b is the depth in the y-axis and c is the height in the z-axis.

Consider X(0,0,x) a point on the height of the ellipsoid such that -c \leq x \leq c. The plane parallel to xOy going through the point X 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.

1/ellipsoidcap2-746.png
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Example 1

#include <stdio.h>
#include <codecogs/maths/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

Note

the value of x must satisfy the inequality -c \leq x \leq c.

Parameters

xthe coordinate which determines the ellipsoidal cap
athe x semi-axis (width)
bthe y semi-axis (depth)
cthe z semi-axis (height)

Returns

the volume of the ellipsoidal cap

Authors

Eduard Bentea (November 2006)
Source Code

Source code is available when you agree to a GP Licence or buy a Commercial Licence.

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