by David Wise

This investigation serves as introduction to the polar coordinate system and polar equations. We will begin with a discussion of the polar coordinate system and develop an understanding of basic polar equations that produce a few of the "famous" families of curves, specifically spirals, cardioids, and n-leaf roses. There are many other "famous" curves to be explored and each aspect of this investigation could be explored with more depth. This investigation should serve as a good reference tool and spring-board into other investigations.

These demonstration graphs have been produced using the graphing
calculator software, Graphing Calculator 2.2.1. If you do not
have this software, you can use a demo version, or purchase the
software through the internet at **http://www.pacifict.com.**
These demonstrations can also be created using other graphing
software, or graphing calulators.

**The Polar Coordinate System**

Polar equations are based upon the polar coordinate system, which is constructed as follows.

- Begin with point O, the called
**polar origin**or**pole**, and ray emanating from O called the**polar axis**. The polar axis is usually drawn horizontally and extending to the right (just as the posititve portion of the x-axis in the rectangular coordinate system). - Each point P is then labelled with a pair of coordinates
**(r, q)**. **r**is the**directed distance**from O to P and can be any real number.**q**is the**directed angle**from the polar axis to the line segment OP. Follow the trigonometric laws that establish counterclockwise rotations as a positive angle measure and clockwise rotations as a negative angle measure.- A polar point is not unique (as with a rectangular point).
Each polar point has an infinite number of pairs of coordinates,
since adding or subtracting by multiples of 2p
provides rotations to the same terminal side. Meaning,
**(r, q)**and**(r, q + 2kp)**, where k is any integer, are coordinates of the same point. In addition,**(r, q)**and**(-r, q + p)**and**(-r, q - p)**are coordinates of the same point.

The relationship between polar coordinates and rectangular coordinates are as follows.

- To convert from polar to rectangular coordinates, given
**(r, q)**, then**x = r cos q**and**y = r sin q**. - To convert from rectangular to polar coordinates, given
**(x, y)**, then**r**^{2}**= x**^{2}**+ y**^{2 }and**tan q = y/x**.

**A Look at Polar Equations**

**Spirals:**

The** spiral** family of curves is produced by
the general equation r = aq.

The following graph is of the equations:

*a*
controls the "size", or "tightness", and is
the scalar of the spiral. The range of *q* in the graph
represents 1 positive, and therefore, counterclockwise rotation.
As qmax
is increased to +infinity, r increases to +infinity. Adjusting
the range of *q* produces
the following graph:

The graph now includes 1 negative, and
therefore, clockwise rotation. As qmin approaches -infinity,
r approaches -infinity. Negative and positive rotations result
in a "mirror image", or a "flip" about the
vertical axis. Revisiting *a*, when *a* is negative,
the following graph is produced.

The negative produces a rotation by p
(or -p) of the *+a *graph.

To explore the spiral family of curves
more using Graphing Calculator 2.2.1 as a helper application,
**click here**.

**Cardioids:**

The **cardioid** family
of curves are named for their heart shape and are produced by
the following general equations.

__Similarities among cardioids:__

- To obtain any cardioid, the range of q must equal the distance of 2p. The range can include any combination of qmin and qmax, as long as the rotation distance equals 2p. Therefore, it is standard to set qmin = 0 and qmax = 2p.
- The graphs produced by
**r = a(cos****q + 1) and r = a(cos q - 1)**are identical to each other. - The graphs produced by
**r = a(sin****q + 1) and r = a(sin q - 1)**are identical to each other.

The following graph is of the two most basic cardioids.

Replacing cosine with sine results in
a p/4
rotation of the cardioid. Notice the intercepts of each of the
graphs in relation to the corresponding equations. Looking at
just cosine cardioids, the effect of *a* is shown in the
following graph.

*a*
controls the "size" and is the scalar of the cardioid.
Again, notice the intercepts of each of the graphs in relation
to the corresponding equations. The horizontal intercepts are
0 and 2*a* and the veritical intercepts are *a*
and *-a*. *a* has the same effect on sine cardioids,
but remember to take into account the p/4 rotation. Therefore,
the horizontal intercepts are *a* and *-a* and the
vertical intercepts are 0 and 2*a*. If *a* is negative,
the following graph results:

*-a*
produces a "flip" about the vertical axis.* -a*
has the same effect on sine cardioids, but the "flip"
is about the horizontal axis.

To explore the cardioid family of curves
more using Graphing Calculator 2.2.1 as a helper application,
**click here**.

**n-Leaf Roses:**

The **rose** family of curves
are named for their flower shape and are produced by the following
general equations.

__Similarities between rose and cardioid
equations:__

*a*controls the "size" and is the scalar of the rose. However, for roses the horizontal and/or vertical intercepts are at 0 and*a*.- Replacing cosine with sine results in a p/4 rotation of the rose. Therefore, all horizontal intercepts become vertical intercepts and all vertical intercepts become horizontal intercepts.
*-a*produces a "flip" about the vertical axis for cosine roses and a "flip" about the horizontal axis for sine roses.

The following graph is of n-leaf roses, when n is odd.

To obtain an odd n-leafed rose, the range
of q
must equal the distance of p. The range can include any combination of qmin
and qmax,
as long as the rotation distance equals p. Therefore, the
standard default of qmin = 0 and qmax = 2p
graphs the set of all points twice
around the curve. **n = the number of leaves of the rose**.
When n = 1, a special case arises, the 1-leaf rose is a circle.
Notice that the intercepts are at 0 and *a*.

To explore odd-leaf roses more using
Graphing Calculator 2.2.1 as a helper application, **click here**.

The following graph is of n-leaf roses, when n is even.

To obtain an even n-leafed rose, the
range of q must equal the distance of 2p. The range can include
any combination of qmin and qmax, as long as the rotation distance equals
2p.
Therefore, it is standard to set qmin = 0 and qmax
= 2p.
**n = 1/2 the number of leaves of the rose**. Notice
that the intercepts are at 0, *a, *and *-a*.

To explore even-leaf roses more using
Graphing Calculator 2.2.1 as a helper application, **click here**.

If you have any comments concerning this investigation that
would be useful, especially for use at the high school level,
please send e-mail to **esiwdivad@yahoo.com**.

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