Chapter 2, Problem 14P

Two distances are required to specify the location of a point relative to an origin in two-dimensional space (Fig. P2.14):

The horizontal and vertical distances $\left(x, y\right)$ inCartesian coordinates

The radius and angle $\left(r,\theta \right)$ in radial coordinates.

FIGURE P2.14

It is relatively straight forward to compute Cartesian coordinates $\left(x,y\right)$ on the basis of polar coordinates $\left(r,\theta \right)$. The reverseprocess is not so simple. The radius can be computed by the following formula:

$r=\sqrt{x^2+y^2}$

If the coordinates lie within the first and fourth coordinates $\left(\text{i}\text{.e}.,x>0\right)$, then a simple formula can be used to compute $\theta$

$\theta ={\tan }^{-1}\left(\frac{y}{x}\right)$

The difficulty arises for the other cases. The following table summarizes the possibilities:

x	Y	$\theta$
$\text{<}0$	$\text{>0}$	${\tan }^{-1}\left(y/x\right)+\pi$
$\text{<}0$	$\text{<}0$	${\tan }^{-1}\left(y/x\right)-\pi$
$\text{<}0$	$=0$	$\pi$
$=0$	$\text{>0}$	$\pi /2$
$=0$	$\text{<}0$	$-\pi /2$
$=0$	$=0$	0

(a) Write a well-structured flowchart for a subroutine procedure to calculate r and $\theta$ as a function of x and y. Express the finalresults for $\theta$ in degrees.

(b) Write a well-structured function procedure based on your flowchart. Test your program by using it to fill out the following table:

x	Y	r	$\theta$
1	0
1	1
0	1
–1	1
–1	0
–1	–1
0	–1
1	–1
0	0

To determine

A well-structured flowchart for a subordinate procedure to calculate r and $\theta$ as a function of x and y for the table given below, if two distances are required to specify the location of a point relative to an origin in two-dimensional space.

$\begin{array}{ccc}x& y& \theta \\ <0& >0& {\tan }^{-1}\left(y/x\right)+\pi \\ <0& <0& {\tan }^{-1}\left(y/x\right)-\pi \\ <0& =0& \pi \\ =0& >0& \pi /2\\ =0& <0& -\pi /2\\ =0& =0& 0\end{array}$

Answer to Problem 14P

Solution:

Explanation of Solution

Given Information:

To specify the location of a point relative to origin, cartesian coordinates $\left(x,y\right)$ and polar coordinates $\left(r,\theta \right)$ are used.

Cartesian coordinates can be easily computed from the polar coordinates.

Polar coordinates are computed as follows:

$r=\sqrt{x^2+y^2},\theta =ta{n}^{-1}\left(\frac{y}{x}\right)$

Formula used:

Calculation:

The flowchart for converting from Cartesian to polar coordinates is as follows

To determine

A well-structured function procedure based on flowchart of part (a) if two distances are required to specify the location of a point relative to an origin in two-dimensional space. Also, test the program by filling the table given below:

$\begin{array}{cccc}x& y& r& \theta \\ 1& 0& & \\ 1& 1& & \\ 0& 1& & \\ -1& 1& & \\ -1& 0& & \\ -1& -1& & \\ 0& -1& & \\ 1& -1& & \\ 0& 0& & \end{array}$

Answer to Problem 14P

Solution:

The values of r and $\theta$ corresponding to provided x and y are:

$\begin{array}{cccc}x& y& r& \theta \\ 1& 0& 1& 0\\ 1& 1& 1.4142& 45\\ 0& 1& 1& 90\\ -1& 1& 1.4142& 135\\ -1& 0& 1& 180\\ -1& -1& 1.4142& -135\\ 0& -1& 1& 1\\ 1& -1& 1.4142& -45\\ 0& 0& 0& 0\end{array}$

Explanation of Solution

Given Information:

To specify the location of a point relative to origin, cartesian coordinates $\left(x,y\right)$ and polar coordinates $\left(r,\theta \right)$ are used.

Cartesian coordinates can be easily computed from the polar coordinates.

Polar coordinates are computed as follows:

$r=\sqrt{x^2+y^2},\theta =ta{n}^{-1}\left(\frac{y}{x}\right)$

Calculation:

The MATLAB program for converting Cartesian to polar coordinates is as follows:

% Define a function polar()

function polar(x, y)

% Formula to calculate r

r = sqrt (x.^ 2+ y.^ 2);

if x > 0

% Formula to calculate theta

th= atan(y/ x);

elseif x < 0

% Use for loop to check the condition

if y > 0

th= atan(y / x)+ pi;

elseif y < 0

th= atan(y / x)- pi;

else

th= pi;

end

else

if y > 0

th= pi / 2;

elseif y < 0

th=- pi / 2;

else

th= 0;

end

end

theta = th* 180 / pi;

% Display the polar coordinates

fprintf('r = %4.4f theta = %4.2f\n',r,theta);

Now, to test the program use the following command.

First find polar coordinates for $x=1,y=0$.

OUTPUT:

Now, for polar coordinates for $x=1,y=1$.

OUTPUT:

Now, for polar coordinates for $x=0,y=1$.

OUTPUT:

Now, forpolar coordinates for $x=-1,y=1$.

OUTPUT:

Now, forpolar coordinates for $x=-1,y=0$.

OUTPUT:

Now, forpolar coordinates for $x=-1,y=-1$.

OUTPUT:

Now, forpolar coordinates for $x=0,y=-1$.

OUTPUT:

Now, forpolar coordinates for $x=1,y=-1$.

OUTPUT:

Now, forpolar coordinates for $x=0,y=0$.

OUTPUT:

Hence, the values of r and $\theta$ corresponding to provided x and y are:

$\begin{array}{cccc}x& y& r& \theta \\ 1& 0& 1& 0\\ 1& 1& 1.4142& 45\\ 0& 1& 1& 90\\ -1& 1& 1.4142& 135\\ -1& 0& 1& 180\\ -1& -1& 1.4142& -135\\ 0& -1& 1& 1\\ 1& -1& 1.4142& -45\\ 0& 0& 0& 0\end{array}$