EMAT 4680/6680 Explorations 11

Polar Equations

Last modified on

January 6, 2016

Graphing Calculator 3.5 or 4.0, Geogebra, or Desmosis suggested for these investigations. Some of them could be done with a TI-83 or similar hand-held technology. Or, for a challenge, try making the graphs with Excel.

1. Investigate

Note:

- When a and b are equal, and k is an integer, this is one textbook version of the " n-leaf rose."
- Compare with

for various k. What if . . . cos( ) is replaced with sin( )?

2. Investigate -- varying a, b, c, and k.

Note: The parameter k is called the "eccentricity" of these conics. It is usually called "e" but for many software programs e is a constant and can not be set as a variable.

For notes on a derivation of these formulas, click

here.

4. Investigate

5. Some interesting graphs --

when

aissmall,e.g. a = .1for suitable a and range of

Replace the '10' with n and animate for some appropriate range.

(Note: some of these graphs may become more interesting -- or disasters -- when is replaced by some function of .)

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