This is the 1st part of my write-up of Assignment #11 Brian R. Lawler EMAT 6680 12/14/00

## Polar Equations

The Problem

 Part I. Part II. Part III. Investigate varying a, b, c, and k.   Analysis - Part I.

The discussion and summary of observations assumes the polar coordinate system. Recall that any ordered pairs (m, n) are first a radius (or length) than a clockwise rotation from the positive x-axis. Click on any graph or equation to view and manipulate the function in Graphing Calculator 3.0.
 The graph of with a = 1 and k = 1. Notice this defines a radius 1 circle centered at (1,pi/2).  The graph of with a = 1 and k = 2. This is a 4 petal flower centered at the origin with petals that appear to be 2 in length along their line of symmetry.  The graph of with a = 1 and k = +/- 3. This is a 3 petal flower centered at the origin with petals that appear to be 2 in length along their line of symmetry. Notice the negative inverts the original flower.   Continuing to change the values for k will just continue to add petals to the flowers observed. When k is an even number, there are always 2k as many petals. The first petal counterclockwise from the positive x-axis is at (2pi / 4k) radians and the next petals are (2pi / 2k) radians counterclockwise from there. When k is an odd number, there are k petals. One petal will be on the negative y axis, with the rest (2pi / k) radians counterclockwise from there. A negative value for k only has an observable effect on odd values of k, as seen in the final example above. This flips the image of the graph with positive k over the horizontal axis. Next to look at the graphs of the form . First consider when a = 1 and k = 1. Notice this defines a radius 1 circle centered at (1, 0). As you may expect, there is a clockwise rotations of pi/2 compared with the same function that replaces the cosine with sine (see the green circle above).  The graph of with a = 1 and k = 2. This is a 4 petal flower centered at the origin with petals that appear to be 2 in length along their line of symmetry. Again, there is a clockwise rotations of pi/2 compared with the same function that replaces the cosine with sine (see the yellow flower above).  The graph of with a = 1 and k = +/- 3. This is a 3 petal flower centered at the origin with petals that appear to be 2 in length along their line of symmetry. Again, there is a clockwise rotations of pi/2 compared with the same function that replaces the cosine with sine. However, notice that both graphs do not appear. In fact, both are present and the same inversion about the x-axis occurs as seen in the sine function above. However, because of the pi/2 rotation, this inversion maps the flower right back to itself.   Continuing to change the values for k will just continue to add petals to the flowers. This behavior observed in the cosine functions summarily acts the same as the sine examples above. When k is an even number, there are always 2k as many petals. The first petal counterclockwise from the positive x-axis is at 0 radians of counterclockwise rotation and the next petals are (2pi / 2k) radians counterclockwise from there. When k is an odd number, there are k petals. One petal will be on the positive x axis, with the rest (2pi / k) radians counterclockwise from there. A negative value for k only has no observable effect on values of k, as seen in the final example above.

 The a value adjusts that radius of these graphs. In fact, as written, each petal has a length of 2a. Notice that the circles that are generated when k = 1 have also have a diameter of 2a. At right is an animation of varying values for a. Here you can observe the effects of changing a. Click on the equation below to investigate further.   (an aside: I was curious to make sure my independent investigation of the two variables was complete. In other words, I wanted to make sure that I didn't miss something that might happen as both variables moved together. To check this, I ran a few more examples. I did not find anything behaving differently than described above.) click to continue to Part II. Comments? Questions? e-mail me at blawler@coe.uga.edu

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