A point in polar co-ordinates is represented as (r, theta). Here, r is its distance from the origin and theta is the angle at which r has to be measured from origin. Any mathematical function in the Cartesian coordinate system can also be plotted using the polar coordinates.
Modules required
- Matplotlib : Matplotlib is a comprehensive Python library for creating static and interactive plots and visualisations. To install this module type the below command in the terminal.
pip install matplotlib
- Numpy : Numpy is the core library for array computing in Python. To install this module type the below command in the terminal.
pip install numpy
- math : math is a built-in module used for performing various mathematical tasks.
The matplotlib.pyplot module contains a function polar(), which can be used for plotting curves in polar coordinates.
Syntax : matplotlib.pyplot.polar(theta, r, **kwargs)
Parameters :
- theta – angle
- r – distance
Approach :
In each of the examples below,
- A list of radian values is created. These values cover the domain of the respective function.
- For each radian value, theta, a corresponding value of r is calculated according to a specific formula for each curve.
1. Circle : A circle is a shape consisting of all points in a plane that are a given distance(radius) from a given point, the centre. Hence, r is a constant value equal to the radius.
Example :
Python3
import numpy as npimport matplotlib.pyplot as plt # setting the axes projection as polarplt.axes(projection = 'polar') # setting the radiusr = 2 # creating an array containing the# radian valuesrads = np.arange(0, (2 * np.pi), 0.01) # plotting the circlefor rad in rads: plt.polar(rad, r, 'g.') # display the Polar plotplt.show() |
Output :
2. Ellipse : An ellipse is the locus of a point moving in a plane such that the sum of its distances from two other points (the foci) is constant. Here, r is defined as :
Where,
- a = length of semi major axis
- b = length of semi minor axis
Example :
Python3
import numpy as npimport matplotlib.pyplot as pltimport math # setting the axes# projection as polarplt.axes(projection = 'polar') # setting the values of# semi-major and# semi-minor axesa = 4b = 3 # creating an array# containing the radian valuesrads = np.arange(0, (2 * np.pi), 0.01) # plotting the ellipsefor rad in rads: r = (a*b)/math.sqrt((a*np.sin(rad))**2 + (b*np.cos(rad))**2) plt.polar(rad, r, 'g.') # display the polar plotplt.show() |
Output :
3. Cardioid : A cardioid is the locus of a point on the circumference of a circle as it rolls around another identical circle. Here, r is defined as :
Where, a = length of axis of cardioid
Example :
Python3
import numpy as npimport matplotlib.pyplot as pltimport math # setting the axes# projection as polarplt.axes(projection = 'polar') # setting the length of # axis of cardioida=4 # creating an array# containing the radian valuesrads = np.arange(0, (2 * np.pi), 0.01) # plotting the cardioidfor rad in rads: r = a + (a*np.cos(rad)) plt.polar(rad,r,'g.') # display the polar plotplt.show() |
Output :
4. Archimedean spiral : An Archimedean spiral is the locus of a point moving uniformly on a straight line, which itself is turning uniformly about one of its end points. Here, r is defined as :
Example:
Python3
import numpy as npimport matplotlib.pyplot as plt # setting the axes# projection as polarplt.axes(projection = 'polar') # creating an array# containing the radian valuesrads = np.arange(0, 2 * np.pi, 0.001) # plotting the spiralfor rad in rads: r = rad plt.polar(rad, r, 'g.') # display the polar plotplt.show() |
Output :
5. Rhodonea : A Rhodonea or Rose curve is a rose-shaped sinusoid plotted in polar coordinates. Here, r is defined as :
Where,
- a = length of petals
- n = number of petals
Example:
Python3
import numpy as npimport matplotlib.pyplot as plt # setting the axes# projection as polarplt.axes(projection='polar') # setting the length# and number of petalsa = 1n = 6 # creating an array# containing the radian valuesrads = np.arange(0, 2 * np.pi, 0.001) # plotting the rosefor rad in rads: r = a * np.cos(n*rad) plt.polar(rad, r, 'g.') # display the polar plotplt.show() |
Output :
