Hypocycloids and Hypotrochoids

Roulettes – 4: Hypocycloids and Hypotrochoids In our last few posts we investigated rouletts, the curves that are formed by the locus of points attached to a circle as it rolls around the outside of a fixed circle. Depending on the ratio of the radii (and therefore the circumferences) of the circles these curves are the…

Epitrochoids

Roulettes – 3: Epitrochoids In the first post of this series Roulette Generators are explained. Here are the files for Winplot or Geometer’s Sketchpad. Use them to quickly see the graphs of these curves by adjusting one or two parameters. The parametric equations of these curves are below. R and S are parameters that are adjusted for each curve. Before looking at Epitrochoids, consider the…

Epicycloids

Roulettes – 2: Epicycloids In the last post we saw how a cardioid can be generated by watching the locus of a point on as one circle rolls around another circle with the same radius. In the first post of this series Roulette Generators are explained. Here are the files for Winplot or Geometer’s Sketchpad. Use them to quickly see the graphs of…

Rolling Circles

A few weeks ago I covered some trigonometry classes for another teacher. They were studying polar and parametric graphs and the common curves limaçon, rose curves, cardioids, etc. I got to thinking about these curves. the next few posts will discuss what I learned. To help me see what was happening I made a Winplot…

New Look

            I am back from my sojourn in Hawai’i. It was a great year. I enjoyed being back in the classroom and working with the kids. My wife and I had a whale of a good time. I decided to freshen up the look of the blog with a new…

A Problem with 4 Solutions and 2 Morals

A friend of mine e-mailed me yesterday with a question: Her class was using the Law of Cosines and came up with a solution of , but the “correct” answer was . She wanted to know, having gotten the first answer, how do you get to the second from it. The answers are equivalent: I figured it…

My Favorite Function

My favorite function is . I like to ask folks how many zeros this function has on the interval . Most folks will get their calculator out and graph the function on the interval Two zeros: one at 1 and the other about 0.05 more or less. So then I suggest they look at .…