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Hello Constance and the Kevins,

This is Xintong, from Swarthmore College. Hope you all are having a fun and relaxing summer. You all have done some good work on the page so far, and I was especially impressed at how you guys learned parametric equations and polar coordinates so quickly. I have a few suggestions for the page.

Image and Image Description (at the very top)
  • I think you should somehow combine the two sentences. You try to use roulette in the first sentence to help describe epitrochoids, but you define a roulette in the second sentence. This could cause unnecessary pauses for readers.
Basic Description
  • Instead of “The basic concept of this is:...”, write “An epitrochoid is made up of…”
  • Give names to the circle that can’t move and to the circle that can (something like Circle A and Circle B will work).
  • The 2nd sentence can be removed; if a circle can’t move, then of course it’s fixed.
  • In the 3rd sentence, what does “a line that will create a line” mean? And a line can’t create anything.
  • I don’t really understand the rabbit example. I think a ball rolling around the earth without slipping would be better. A rabbit just has no sense of rolling.
A More Mathematical Explanation
While I commend your efforts in trying to understand the derivation of the parametric equation, I think it’s a bit out of reach. Unless you can prove to me that you fully understand the derivation, please remove the entire parametrization. Trying to explain something that you don’t understand just doesn’t make much sense. Of course, this means removing all the velocity, acceleration, arc length, and curvature equations as well because they’re even harder to comprehend.
Instead, I wish for you all to focus on the geometric elements of epitrochoids, such as how the ratio of the two circles affects how many times the outer circle goes around the inner circle. I know there is already some work on this, but you can definitely make it more rigorous and expand it further. If you need more suggestions on this, let me know.
More general suggestions:
  • Be sure to use math environment ( math here (look at edit for code) ) to write longer equations. Variables in a line of text should be italicized instead of quoted.
Special Epitrochoids (within the Math section)
  • Defining a Limaçon was just out of the blue. Try to include the image of it right next to it instead of with the other 3 epicycloids.
  • Try to describe the image of the three cases comparing b to a. Without the parametric equations, try to logic through why each case of b vs. a leads to the particular image.
  • On the image with the 4 epicycloids, label each one or it’s pretty confusing.
Explanation on the Position of the Point
  • Make the first image within this section slightly smaller.
  • In the text right under the image mentioned above, say “The location of the point does not affect…” instead of “Where the point is does not affect…”.
Relationship Between Radii and Circumference with the Number of Bumps
  • In your first test, you talk about how 3/1 being a whole number means the curves stay at the same place. However, you never derived that fact, so it doesn’t make much sense to state that.
  • I suggest actually deriving the relationship between the ratio of the radiuses of the two circles and the amount of rotations the outer circle will do in a single revolution. Given that, you can conclusively prove your “bumps” theory too, instead of just looking at a few tests and giving generalizations (can’t prove something with examples in rigorous mathematics). Let me know if you need some help on this. This also helps with looking at your complicated epitrochoids.
That’s it for now in the Math section. After more work gets done, I’ll write another feedback.
Why It’s Interesting
You have a good start to the Wankel Rotary, the Spirograph, and the Ptolemaic System. However, you can definitely expand it far more to make it even more interesting. For example, you mention how epitrochoids have a role in all 3 of these things, but never mention specifically how so.

Alright that’s it for now. Great job on the page so far, and try to still enjoy your summers while you work on the page!

--Xintong Tian 18:08, 3 July 2013 (EDT)