Polar Equations

by Oktay Mercimek

Let's start to investigate with first one.

  for a=1 and k=1 , equation is  which is a circle equation in polar coordinates.

Making a bigger doesn't change that much , it just changes the size of circle.

Graph for

If we set a as negative number, it changes the direction of circle

For

For k>1 , graph is changing slightly

If a=1 and k=2, then graph looks like

And again changing a is changes the size of curve

Let's make k=3 and see what is going on

Isn't is little different than what you expected?

make it k=4 and k=5

(k=4)                                    (k=5)

Can we say it is different for odd and even values for k ?

For k=10 , I expect 20 leaves

And for k=11, I expect 11 leaves

To understand the behavior of this function, I put k=n and make n variable between 0 and 10.

So it is  for   , Click here to view Graphing Calculator File.

Click play button to see how it changes while n varies and you will be surprised when you see the difference between odd and even values for k. (you can download Graphing Calculator Viewer (FREEWARE) from http://www.pacifict.com/FreeStuff.html in order to open a GCF file)

has similar to , for k=1 circle is on the x-axis instead of y-axis.

For other k values, it is again similar to sin function but rotating with certain angle

 To understand better you can look HERE where a varies  (ASSGN11 A vary) or HERE for k varies.

More amazing things is happening with 

Let's look for couple b values,

For b=05

for b=1.2

for b=2

for b=3

Click HERE to GCF (COS B VARY) file for other b values for  function .

For sin version click HERE  (SIN B VARY).

As a last investigation let's look this function 

Let's make c=10, a=1 , b=1 and k=4

   Isn't it nice?

click Here for different c values  ( LAST C VARY)

For c=10, a=1 , b=1 and k=5.5

click Here for different k values  ( LAST K VARY)

click Here for different a values  ( LAST A VARY)

click Here for different b values  ( LAST B VARY)

You can see that changing k is related with the expanding lines and curves, changing c is related with magnifying picture of graph, and changing a and b is related with rotating the picture of graph.