math.gif (3247 bytes)POLAR EQUATIONS AND THEIR GRAPHS

Copyright by Ingrid Stewart, Ph.D.  Please Send Questions and Comments to ingrid.stewart@csn.edu.

It is often necessary to transform from rectangular to polar form or vice versa.  The following polar-rectangular relationships are useful in this regard.

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Strategy for Changing Equations in Rectangular Form to Polar Form

Strategy for Changing Equations in Polar Form to Rectangular Form

In previous lecture notes you spent a lot of time learning how to graph equations in the Rectangular Coordinate System by hand.  Polar equations can also be graphed by hand in the Polar Coordinate System, however, this task often becomes extremely difficult.  That is why we try to use graphing calculators or computer programs to help us visualize the graphs of polar equations. 

For the following "special" polar equations, you need to be able to associate their name with their characteristics and graphs!

Equations of Circles

Circles with center along one of the coordinate axes and radius a!

              Circle with center at the origin and radius a!

NOTE:  a is NOT equal to 0

The graphs of Circles are generated as the angle increases from 0 to 2pi.gif (853 bytes).

a.   (equation of a Circle with center at (1,0) and radius 1)

b.   (equation of a Circle with center at (0,0) and radius 2)

Equations of Limacons

, where a and b are NOT equal to 0

The graphs of Limacons are generated as the angle increases from 0 to 2pi.gif (853 bytes).

a.   Limacon with inner loop:

Note that a = 2 and b = 3 and

b.   Dimpled Limacon:

Note that a = 3 and b = 2 and

c.   Convex Limacon: .   Note that the graph is not quite circular!

Note that a = 8 and b = 2 and

Equations of Cardioids

       , where

The graphs of Cardioids are generated as the angle increases from 0 to 2pi.gif (853 bytes).

(equation of a Cardioid)

Note that a = 2 and b = 2 and

Equations of Rose Curves

, where a and n are NOT equal to 0

a.   (equation of a Rose Curve with 3 petals)

Note that n = 3 is odd, therefore the rose curve has 3 petals.

b.   (equation of a Rose Curve with 8 petals)

Note that n = 4 is even, therefore the rose curve has 2(4) = 8 petals.

Equations of Lemniscates

, where a is NOT equal to 0

The graphs of Lemniscates are generated as the angle increases from 0 to 2pi.gif (853 bytes).

( equation of a Lemniscate)

In order to graph a lemniscate with a calculator or a computer program we must solve its equation as r in terms of theta.gif (860 bytes).  Therefore,

and .

We will get the same graph no matter which one of the two equations we graph!

         

Most polar functions are best graphed with a graphing utility because one usually requires many points in order to find the characteristics of each graph. 

Example 01
Change a rectangular equation to polar form.
Example 02
Change a rectangular equation to polar form.
Example 03
Change a rectangular equation to polar form.
Example 04
Change a rectangular equation to polar form.
Examples 05 and 06
Change polar equations to rectangular form.
Examples 07 and 08
Change polar equations to rectangular form.
Example 09
Change polar equations to rectangular form.
Example 10
Change polar equations to rectangular form.
Example 11
Change polar equations to rectangular form.
Example 12
Change polar equations to rectangular form.
Example 13
Change polar equations to rectangular form.
Example 14
Change polar equations to rectangular form.