properties of

This exploration will focus on effects of a,b,n, and on the following polar equations:

and

Since both equations exhibit similar properties, will mainly be used as the source for this exploration. To understand the effects of each component (a,b,n, and ), we can strip our original equation to the base equation: a simple graph of circle centered at (0.5,0)

The first observation that can be made is, if we alter our base equation and add a varying n as a coefficient of we change the number of nodal points of the graph. For instance the following graphs at different n's for :

a)b)
`c)d)`

Challenge: Try and match the graphs with the corresponding n values

n = 3 (a) (b) (c) (d)

n = 4 (a) (b) (c) (d)

n = 6 (a) (b) (c) (d)

n = 9 (a) (b) (c) (d)

To see a movie of varying the n value click here ->

The second observation that can be made deals with the coefficient b multiplied to the polar equation. Giving us the equation:

Similar to the amplitude of a cos or sin function in rectangular coordinates, the b coefficent either increases or decreases the diameter/length of the circle/propellers. For the following graphs, try your intution again and see if you can predict which b value will go with the correct graph.

a)b)

c) d)

n = 1, b = 5 (a) (b) (c) (d)

n = 3, b = 2 (a) (b) (c) (d)

n = 1, b = 2 (a) (b) (c) (d)

n = 3, b = 5 (a) (b) (c) (d)

To see a movie of varying the b value at n=3 click here ->

Lastly, we will add a constant a to our newly constructed polar equation giving us the final equation of->

As opposed to the previous observations, adding the a constant a will drastically change the behavior of the graph. By varying the constant a, you will notice that the eccentricty of the graph decreases, becoming more and more like a polygon. For higher values of a, you will have a trend toward convex polygons as opposed to non-convex shapes. For an example click here for a movie illustrating this observation.

When comparing, sin and cos polar equations, you need to keep in mind the behavior of sin and cos in the rectangular coordinate system. With that in mind, the behavior of sin and cos in the polar coordinate system are very similar. For the following equations:

purple and red

A simple observation would be that the sin graph is just a counter-clockwise rotation of the cos graph ninety-degrees. This can further be validated by varying the n for both graphs and getting a pinwheel-like behavior click here-> for the movie.