5. Doing math with Numeric Arrays

In this chapter, we will see how to plot curves using Numeric arrays and functions like plot and polar.

5.1. Plotting a sine curve

Say you wish to plot one full cycle of the sine curve. The first step is to generate a set of numbers between 0 and 2*pi. This can be done very easily using a function called linspace:

>>> from pylab import *
>>> a = linspace(0, 2*pi, 4)
>>>> a
array([ 0.        ,  2.0943951 ,  4.1887902 ,  6.28318531])
>>> sin(a)
array([  0.00000000e+00,   8.66025404e-01,  -8.66025404e-01,
     -2.44921271e-16])
>>>

The function:

linspace(x, y, N)

generates N equally spaced numbers starting from x and ending in y; the difference between adjacent numbers is fixed. In the above case, x is 0, y is 6.28 and N is 4. The output which linspace generates is a Numeric Array:

array([0, 2.0943951 , 4.1887902, 6.28318531])

You observe that the difference between consecutive numbers is fixed, approximately equal to 2.09. When you apply the sin function on a, it computes sine of all the numbers in the numeric sequence.

Plotting a sine curve with just 4 points is not a good idea; especially when it is so easy to generate any number of points:

>>> x = linspace(0, 2*pi, 200)
>>> y = sin(x)
>>> plot(x, y)
[<matplotlib.lines.Line2D object at 0x9bf4bac>]
>>> show()
>>>

Now, we are generating 200 points (in x) and storing the sin of each of these points in y. The plot function works like this: it takes the first number in x and the first number in y and plots a point, then it takes the second number in x and the second number in y and plots another point ... and so on. Note that when you apply plot, it does not immediately show you the graph - it just prints a line in the output (we need not understand what that line means). Only when the show function is applied do we actually get the plot on the screen!

5.2. Plotting the polar rose

In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point in a plane is determined by a pair of values:

where r represents the distance from a fixed point (called the pole) and theta represents an angle with respect to a fixed direction.

It is easy to plot in polar coordinates, as the following example illustrates.

The polar rose is defined as:

We get a k-petaled rose if k is odd and 2k petaled rose if k is even. Here is a Python code fragment which plots the polar rose:

>>> from pylab import *
>>> theta = linspace(0, 2*pi, 200)
>>> r = 4 * cos(8 * theta)
>>> polar(theta, r)
[<matplotlib.lines.Line2D object at 0xa3efb4c>]
>>> show()
>>>

The function polar works with two numeric arrays - the second one gives the r values of each point in the plane and the first one gives the corresponding angles. Here is the plot:

5.3. Labels and title

You can give a title to your plot by using a function called title (call this function before calling show):

>>> title('A simple Sin curve')
<matplotlib.text.Text object at 0x8d31bec>
>>>show()

Labels can be given to the X and Y axes:

>>> xlabel('X axis')
<matplotlib.text.Text object at 0x89d26ec>
>>> ylabel('Y axis')
<matplotlib.text.Text object at 0x8a87b8c>
>>> show()

xticks and yticks are two other interesting functions. Try out the experiment below to understand how they work:

>>> x = linspace(0, 2*pi, 200)
>>> y = sin(x)
>>> m = arange(0, 2*pi, .4)
>>> n = arange(-1, 1, .2)
>>> xticks(m)
>>> yticks(n)
>>> plot(x, y)
>>> show()

Note

The function call arange(0, 2*pi, 0.4) returns a numeric sequence from 0 to 2*pi - each number in the sequence differs from the next by 0.4.

5.4. Exercises

1. Refer your maths textbook and find out the polar equation of the curve called a cardioid. Try to plot the cardioid using Python.
2. Try to plot a circle in polar coordinates.