Chapter 11.1 Puzzle

### This assignment is due 48 hours after the Chapter Test is given.

• Puzzle solutions are due by 5:00 PM two days after the Chapter Test.
• Puzzles are worth 10 points of extra credit each.
• Answers must be accompanied by valid reasoning. Just like the tests, the answer alone isn't enough!

"Families of Hypocycloids"

(You won't find these at the zoo.)

There is an interesting history behind epicycloids. After Copernicus showed that the sun didn't move around the Earth, astronomers believed that the planets moved in circular paths around the Sun. Gradually, mathematical analysis showed that this wasn't quite right. So, they posited that the "circular paths" were actually epicycles: small circles rolling around larger ones (See Stewart, p. 682). More accurate numerical data showed that this theory was also wrong. It was then believed that the paths were double-epicycles: circles rolling around circles rolling around circles. Finally, Kepler (using Brahe's data) showed that the paths were elliptical, and then Newton, using his newly developed calculus, derived laws to show why Kepler's discovery was true.

If a circle C rolls on the outside of the fixed circle, the curve traced out by point P (a point on the circle C) is called an epicycloid. Find the parametric equation for the epicycloid. (See Stewart, p. 682, for additional information about hypocycloids)