Experimenting with Hypocycloids (Diversion VI)

I fascinated and treated myself with the simple trigonometric functions that I was working with previously just to relieve my stress after having finished a rigorous monthly data summary and report back from work. I never knew someone from Browny-knows-where reminded me of the very entertaining spirograph! Yes, a spirograph - it comprises a set of small, plastic-made wheels of different sizes and two large rings with teeth on their inside as well as outside rims which anyone can basically use to draw families of hypocycloids and epicycloids. The toy was introduced in the western culture in 1970’s; and, though I live in a third world country, I was fortunate enough to have a grasp on this wonderfully educational graphing toy - I quite remember my elementary public school library had me borrow a set one time. 

In this photo, I constructed hypocycloids by using the same smoothed-line XY-scatter plots easily available from Excel. The equations are easy and how selfish would I be if I don’t post them here?!

OK, here they are:

x(Ɵ) = r(k-1)cosƟ + rcos((k-1)Ɵ); and

y(Ɵ) = r(k-1)sinƟ - rsin((k-1)Ɵ).

I decided r to be equivalent to 1 just for ease of encoding and let k be any number (it’s actually the ratio between the radius of the bigger wheel against the smaller one). For the case of my Diversion VI photo, I simply selected the values 7.2, 4, 2.1, and 6.

Now, I’m off to the market to find myself one of those spirographs. Excel must make the plotting easy, but the real toy offers more. 

©karlobenson