**GRAPHS OF **** SOME INTERESTING**** POLAR EQUATIONS
Copyright by Ingrid Stewart, Ph.D. Please Send Questions and Comments to
ingrid.stewart@csn.edu.**

Learning Objectives - This is what you must know after studying the lecture and doing the practice problems!

1. Memorize the general polar forms of circles and associate them with their graphs.

2. Find the radius of a circle equation in polar form.

3. Associate the names of limaçons with their graphs.

4. Recognize the graphs of Rose Curves given their polar form and find the number of petals.

Many polar equations have graphs we already studied in algebra (e.g., lines, parabolas). In this lesson we will discuss a few polar equations whose graphs we did not encounter in algebra. However, we will also revisit circles which we already studied in algebra, but this time their equations will be in polar form.

LimaçonsPronounced lim-uh-sons which is French for "snail". The ç (pronounced c cedilla) is a special French letter. It is a "c" with a hook (cedilla) on the bottom.

where

andaare greater thanb0There are four different types of Limaçons. Their location in the polar coordinate system depends on whether

sineorcosineare present.

You must memorize their pictures together with their namesConvex Limaçon when . It is almost a circle!

Dimpled Limaçon when.

Heart-Shaped Limaçon also called a Cardioid when . As compared to the Dimpled Limaçon, the "dimple" is more pronounced in the Cardioid.

Limaçon with inner loop when .

Rose CurvesTheir equations are

, where

andaare NOT equal tonand0is NOT equal to 1nFollowing are examples of graphs of rose curves. Their location in the polar coordinate system depends on whether

sineorcosineis present.

You must know the pictures of Rose Curves.The number of petals depends on whether the angle multiplier

is odd or even.nWhen

is odd:nThe rose curve has

petals.nYou must memorize this.When

is even:nThe rose curve as

petals.2nYou must memorize this.

Circle Equations in Polar Form

You must memorize their equations.

,r = a sina > 0(See #1in the "Examples" document for Polar Equations.)The graphs of

have their center somewhere along a verticalr = a sinr-axis. The radius is .You must know this.

Example:

r = 3 sinequation of a circle with center atand radius(0. 1.5).1.5

,r = a cosa > 0(See #5in the "Examples" documentfor Polar Equations.)The graphs of

have their center somewhere along the polar axis. The radius is .r = a cosYou must know this.

Example:

equation of a Circle with center atr = 2 cosand radius(1,0).1

,r = aa > 0(See #3 and 6in the "Examples" documentfor Polar Equations.)The graphs of

have their center at the pole. The radius isr = a.aYou must know this.PLEASE NOTE THAT THERE IS NO ANGLE

IN THIS EQUATION. THIS IS UNLIKE THE OTHER CIRCLES, THE LIMAÇONS, AND THE ROSE CURVES. Actually, this means that there is no restriction on. It can take on any value asstays a fixed numberr.a

Example:equation of a Circle with center at

and radius(0,0)2