PolarPlots

2997 days ago by jimlb

Here we give some plots of polar functions.  One can do these by hand by plugging in points and using various symmetries, or one can use a calculator.  Clearly we choose to use SAGE instead.

var('theta') 
       

The following is a rose with 4 petals and maximum radius of 3.  The 2 is controlling the number of leaves.

polar_plot(3*sin(2*theta), (theta,0,2*pi)); 
       

This is what happens when we change the 2 to a 5.

polar_plot(3*sin(5*theta), (theta,0,2*pi)) 
       

And if we change the 2 to a 4..

polar_plot(3*sin(4*theta), (theta,0,2*pi)); 
       

Here are some examples of limacons.

polar_plot(1+2*cos(theta), (theta,0,2*pi)); 
       
polar_plot(3+2*cos(theta), (theta,0,2*pi)); 
       
var('y') 
       

The next graph is of a lemniscate.  The polar equation for the one graphed is $r^2 = 2 \cos(2 \theta)$, but it is easier to graph it in this case using an implicit plot and rectangular coordinates.

P=implicit_plot((x^2+y^2)^2 - 2*(x^2 - y^2), (x, -2, 2), (y, -1, 1)); P.show(aspect_ratio=true)