Great job with the new domain section. I just have a few edits for that section:
- The inner circle generally starts at t=0, so the x coordinate will be a-b+h and the y coordinate will be 0. Similarly, when t is zero, cos(t)=1 and sin(t)=0.
Assuming the second sentence is there to help explain the first sentence, just delete the second sentence. Actually explaining the first sentence would take some additional work that you probably don't have time to do, so I say ditch the explanation completely.
- A full patern will occur when both x and y coordinates have the values of the same initial coordinates at t=0. This occurs at 2\pi, 4\pi, 6\pi: essentially, any multiple of 2\pi.
When what is at any multiple of 2pi? I think you mean when both t is at a multiple of 2pi (so the cos(t) and sin(t) will be 1 and 0 respectively) and also ((a-b)/b)t is a multiple of 2pi, but can you make this explicit?
In your gcd equation, separate 2pi from the fraction b/gcd(a,b), to more clearly reflect the fact that the reader should think of this value as 2pi multiplied by some constant.
Chris Taranta 7/6
I love the image; it makes me think of Spirographs.
Please put in the pronunciation for the word.
Intro: It's hard to distinguish among the three colors.
I recommend defining a hypotrochoid early on. I think the first paragraph under "A More Mathematical Explanation" is accessible even to a non-sophisticated reader and so belongs near or at the top, before "Variations" of a hypotrochoid.
1 Basic Description
- "We will refer to the red point as P" appears in the hide mode but disappears in the show mode.
2 A More Math Exp.
- Are all hypotrochoids graphed using the same two parametric equations? If so, I find that one of the most interesting findings of your wiki. Can you explain why and how?
- Is there a way to determine the domain needed for t in a particular hypotrochoid?
- Please give the values for a, b, and h in the animation.
How the Main Image Relates
- The title is vague: relates to what?
- I would combine the paragraphs and rewrite sentence 3. "What makes them look like three different curves is the fact that only a limited number of sample points are plotted for each graph. Rather than plotting infinitely many points, there are..."
- Why don't the curves with more points completely overwrite the curves with fewer points? Please explain.
This page is really interesting. Most of what you put in the more mathematical section is totally accessible even if you don't know anything about parametrization, so I have a couple of reorganization ideas:
- Move everything except the parametric equations themselves to the basic description. Everything that accompanies the parametric equations can be moved, if reworded slightly.
- Move the definition of a, b, h, etc, to the more mathematical description. It's only needed when you start using equations.
Also, it would be really cool to go into more depth about the emerging patterns that come from rotating the inner circle many times. You talk about this some already, and the applet introduces many more possibilities. You could encourage people to use the applet and try to create a hypotrochoid that requires many full rotations to be "completed" or to create a hypotrochoid that requires very few.
Another thing I would be really interested in (and maybe you would be?), would be any kind of formula relating the inner radius and the outer radius to the number of full rotations required to complete a pattern. My guess is that it has something to do with factors or multiples. This is another thing people could use the applet to play around with.
One other small note. You refer to a "full pattern" in the page and I've been referring to a "complete pattern" just now. This is kind of sloppy on our parts because "pattern" is a vague term. What we actually mean when we talk about reaching a full pattern is allowing the inner wheel to rotate for enough time that the pen returns to exactly the location where it began, which may require more than one trip around the outer wheel.
The animations are in gif format but it would be better if they could be in some other format that allows for viewers to control the speed and when the rolling starts and stops. The second image from the top of More Mathematical Explanation is the most urgent one.
Roulette mouseover: actually the point is neither fixed nor attached to the circle--but how to say it?? Actually, your formal way is pretty good and not confusing, I think.
"Note that the point P can be anywhere in relation to the interior circle." or anywhere on a line through the center of the interior circle??
Nice stuff, Tanya!
Fix after equations, italicize variables, mouseover for domain, "to vary from 0 to 2pi", 3 complete revs therefore ranging from 0 to 6pi
Thanks for sharing your page with me today. I'll be giving more specific feedback soon.