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Welcome to explore Polaaar........Beauty!!!

Assignment 11: Polar Equations

By Nikhat Parveen

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Before we start to investigate the polar equations, let's discuss about polar coordinate system in general.

Polar Coordinate System

To form the polar coordinate system, fix a point O, called the pole (or origin) and construct from O an initial ray called the polar axis as shown in the figure below. Then each point P in the plane can be assigned polar coordinates (r, q) as follows:

1. r = directed distance from O to P

2.q = directed angle, ccw from polar axis to segment OP

In rectangular coordinates, each point (x,y) has a unique representation. This is not true for polar coordinates. Because r is a directed distance, the coordinates  (r, q) and (-r, q + p) represent the same point. In general, the point (r,  q) can be represented as:

(r, q) = (r, q + or - 2np) or (r, q) = (-r, q + or -(2n+1)p), where n is any integer. Pole = (0, q) where  q is any angle.

Also, the polar coordinates (r, q) are related to the rectangular coordinates (x, y) as follows.

x = r cos q ; y = r sin q and tan (q) = y/x;     r2 = x2 + y2

*-*-*-*-*-*+*-*-*+*-*-*-*-*+*-*-*-*-*-*+*-*-*-*-*-*

Now let's investigate  few polar equations  by using graphic calculator and manipulate the values such as k and  q.

For example the polar equation  r = a + b cos (kq ) gives a cardoids and limacon's  for different values of 'b'.

Example 1: a = k = 1 and b = 1,2,3

                                Cardioid (heart- shaped)  (a/b = 1)               Limacon with inner loop  (a/b < 1)

                  

                                                 a= k = b = 1                                              

 As we notice, the shape of the graph  is the same for different values of 'b' but the inner loop is increasing at constant rate. Also you can see that the inner loop and the outer loop intersect at the origin and crosses the x-axis at b - 1 and b + 1 and the outer loop always crosses the y-axis at + or  - 1. To see how the cardioids behave at b = n click here. What happens to Limacons when b approaches zero?

                                            (1< a/b < 2)                                  (a/b > or = 2)

                                        Dimpled Limacon                        Convex Limacon

     

                                                                                                   

The Limacons represent the family of polar equations r = a + (or-) b cos q and r = a + (or -)b sin q where a > 0  and b > 0.

*-*-*-*-*-*+*-*-*+*-*-*-*-*+*-*-*-*-*-*+*-*-*-*-*-*

Our next series of graphs, called rose petals, occur when we keep a and b of our equation, r = a + b cos (kq), constant. For the purposes of the following investigation, we will assign a = b = 1. As we saw above with r = 1 + cos (q), there was only one petal. You will notice when k = 2 there are two petals and, likewise, when k = 3 there are three petals. In fact, when k = n there are n petals. Below are polar equations of the form r = a + b cos (kq) with k = 2, 3, 10, and 1000

       

                                                                                       

 

           

                                                                                                  

Looks like as k increases, not only the number of petals increases but takes a beautiful and artistic form. However, as k increases the visibility of petals decreases. The other fact is that with some higher k values the graphs are simply disaster!!

The polar equations r = 1 + cos (-kq) produces same graph as r = 1 + cos (kq).

 

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To learn more about Cardioids follow the links below:

http://www.cut-the-knot.org/ctk/Cardi.shtml

http://www.xahlee.org/SpecialPlaneCurves_dir/Cardioid_dir/cardioid.html

 

Reference:  "Pre calculus with Limits" A Graphing Approach 3rd Edt., Ron Larson, Robert P.Hostetler, Bruce H.Edwards