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Computes the volume of an ellipsoidal cap.

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Interface

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

using namespace Maths::Geometry::Volume;

inline double ellipsoidal_cap(double x, double a, double b, double c)

This module calculates the volume of an ellipsoidal cap.

The general equation for the ellipsoid is:

$\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$

(1)

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:

$V = \pi a b\left( \frac{2c}{3} - x + \frac{x^3}{3c^2}\right)$

(2)

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

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

Parameters

the coordinate which determines the ellipsoidal cap

the x semi-axis (width)

the y semi-axis (depth)

the z semi-axis (height)

Returns

the volume of the ellipsoidal cap

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

Interactive Calculator

Result –