Chapter 6, Problem 15E

To determine

To write:

A program to convert the Cartesian coordinates to spherical coordinates, and print the results.

Answer to Problem 15E

Solution:

The script file is,

% MATLAB code to print the result in spherical coordinates.

%script file.

[x, y, z] = getcartesian();

%get the value of cartesian coordinates by calling the function

%getcartesian.

printspherical(x, y, z)

%print the value in spherical coordinates by calling the function

%printspherical.

% end of script

%The script file should be placed in the same folder.

The function file is,

% MATLAB code to get the result in cartesian coordinates.

%Function file.

function [x, y, z] = getcartesian()

%get the cartesian coordinates by using the function getcartesian.

x = 1;

%define the variable x.

y = 2;

%define the variable y.

z = 3;

%define the variable z.

end

% end of function

%The function file should be placed in the same folder.

The function file is,

% MATLAB code to print the result in spherical coordinates.

%Function file.

function printspherical(x, y, z)

%the spherical coordinates will be printed by using the function

%printspherical.

[rad, incl, azi] = convert2spher(x, y, z);

%convert the spherical coordinates to cartesian coordionates.

fprintf('the radius is %.2f\n', rad);

%print the radius.

fprintf('the inclination angle is %.2f\n', incl);

%print the inclination angle.

fprintf('the azimuth angle is %.4f\n', azi);

%print the azimuth.

end

% end of function

function [ra, in, a] = convert2spher(x, y, z)

%call a subfunction convert2spher to transform the cartesian coordinates

%into spherical coordinates.

ra = sqrt(x^2+y^2+z^2);

%define the radius.

in = acos(z/ra);

%define the inclination angle.

a = atan(y/z);

%define the azimuth angle.

end

%The function file should be placed in the same folder.

Explanation of Solution

Consider the Cartesian coordinates are,

$\left(x,y,z\right)=\left(1,2,3\right)$

The spherical coordinates are $\left(r,\theta ,\varphi \right)$.

The formulas to convert the Cartesian coordinates to spherical coordinates are as follows.

$\begin{array}{rll}r& =& \sqrt{x^2+y^2+z^2}\\ \theta & =& {\cos }^{-1}\left(\frac{z}{r}\right)\\ \varphi & =& {\tan }^{-1}\left(\frac{y}{x}\right)\end{array}$

Substitute 1 for $x$, 2 for $y$ and 3 for $z$ in the all formulas.

$\begin{array}{rll}r& =& \sqrt{{\left(1\right)}^2+{\left(2\right)}^2+{\left(3\right)}^2}\\ r& =& \sqrt{14}\\ r& =& 3.7417\end{array}$

The inclination angle is,

$\begin{array}{l}\theta ={\cos }^{-1}\left(\frac{3}{3.7417}\right)\\ \theta =36.699°\\ \theta =0.6405\text{rad}\end{array}$

The azimuth angle is,

$\begin{array}{l}\varphi ={\tan }^{-1}\left(\frac{2}{1}\right)\\ \varphi =63.4349°\\ \varphi =1.1071\text{rad}\end{array}$

The spherical coordinates are $\left(3.7417,0.6405,1.1071\right)$.

MATLAB Code:

% MATLAB code to print the result in spherical coordinates.

%script file.

[x, y, z] = getcartesian();

%get the value of cartesian coordinates by calling the function

%getcartesian.

printspherical(x, y, z)

%print the value in spherical coordinates by calling the function

%printspherical.

% end of script

%The script file should be placed in the same folder.

% MATLAB code to get the result in cartesian coordinates.

%Function file.

function [x, y, z] = getcartesian()

%get the cartesian coordinates by using the function getcartesian.

x = 1;

%define the variable x.

y = 2;

%define the variable y.

z = 3;

%define the variable z.

end

% end of function

%The function file should be placed in the same folder.

% MATLAB code to print the result in spherical coordinates.

%Function file.

function printspherical(x, y, z)

%the spherical coordinates will be printed by using the function

%printspherical.

[rad, incl, azi] = convert2spher(x, y, z);

%convert the spherical coordinates to cartesian coordionates.

fprintf('the radius is %.2f\n', rad);

%print the radius.

fprintf('the inclination angle is %.2f\n', incl);

%print the inclination angle.

fprintf('the azimuth angle is %.4f\n', azi);

%print the azimuth.

end

% end of function

function [ra, in, a] = convert2spher(x, y, z)

%call a subfunction convert2spher to transform the cartesian coordinates

%into spherical coordinates.

ra = sqrt(x^2+y^2+z^2);

%define the radius.

in = acos(z/ra);

%define the inclination angle.

a = atan(y/z);

%define the azimuth angle.

end

%The function file should be placed in the same folder.

Save the MATLAB script with name, paracscript.m and function files with names printspherical.m and getcartesian.m in the current folder. Execute the program by typing the script name at the command window to generate result.

Result:

The result is,

Therefore, the result is stated above.