From Math Images
I replaced "The differences between catenaries arises from the scaling factor a from the first equation above" with "The differences between catenaries arises from the scaling factor a in the first equation above," which I'm hoping doesn't feel like stepping on your toes.
I would still say that it is more accurate to say that one equation defines the shape of the parabola, while the second equation just defines one of the symbols in that equation, than it is to that two equations define the catenary (the same way if you have an equation for hours of daylight that involves using the tan function, a second equation defining tan(x) = sin(x)/cos(x) wouldn't be a second equation describing hours of daylight, but would simply define the tan function).
However, this change isn't critical and the page as a whole is excellent. Really interesting, well laid out, good choice of details, well explained, etc. I vote ready.
If you want help figuring out how to say whatever you're getting at with that sentence about the family, we could discuss it in person and you could tell me what you mean, though you could certainly drop the line as well. Whichever you prefer.
My comment from before about how the hidden image in the basic description shouldn't be hidden still holds, especially because the text that explains the image is not hidden, so it's weird to have visible text describing a hidden image.
I still think that what you're trying to say is valuable. I also suggest making my change to the math that I suggested and Abram agreed with. I think it clarifies the equation, though you may want to break it up into two lines (I've seen you align equations elsewhere, so I know you know how to do it).
I understand the issue with saying family. I believe then that it would just be easier for me to take out that sentence and leave it at that...?
Great job on the catenary/parabola section. The wording you chose is much better than my suggested wording. The only thing there I would change would be to unhide the hidden image. It's a useful image, and I think having it shown from the beginning does not clutter the page.
You are absolutely right about your definition of cosh(x), that cosh(x) is defined as (e^x + e^-x) / 2 and nothing else. For this reason, it would be totally correct to say that "cosh" represents the same thing in both equations. However, the entire equation y = a cosh(x/a) is not identical or equivalent to the equation cosh(x) = (e^x + e^-x) / 2. For instance, one equation has a's in it and one doesn't. Anna's recommended fix (below) is a good one.
Thanks for explaining what you mean about the family of curves business. Anna is right that the each possible value of a gives a different curve, and all these curves together make up a family.
Also, I'm not quite sure what you mean by "there are no other equations derived from the hyperbolic cosine." After all, you can make up any old equation you want to using cosh. How about y = cosh(x)^3 + 2* cosh(x/4), or anything else. Are you trying to say that hypberbolic cosine only has one definition? If so, I think you already convey that with the equation that defines cosh as e^x + e^-x / 2. Or maybe there's something else you are getting at?
I had the three part equality before, but Steve told me to change it to how it is now... I don't mind changing it back but just letting you know why it is that way.
Now that I see Abram's comment, I agree that you're using the term "family" in an inappropriate way. I'd say that and are a part of the same family of curves, where the family is defined as the set such that a is an element of the real numbers. is a catenary, which is an element of of the family of curves that I just defined.
I'd also rearrange that section now that I'm looking at it... I'd start by saying "We use the hyperbolic cosine function, to write the equation of the catenary:
or something like that.
I changed the catenary/parabola section and added a mouseover (so now it is botha mouseover and a link) to the word parabola. I added a sentence explaining what I meant by not being a family of curves. All my research shows that cosh(x) is equal to the e^x statement, so I am in no position to change it. You can check Wolfram MathWorld, and I also believe Steve was the one that told me to put the equal sign. Let me know what you think.
This page is organized and presented really well, and I really like your choice of material to include. I agree with Anna that "The Catenary and the Parabola Conceptually" can be moved to the basic description. It's a really nice description. I also agree that the word "parabola" has the potential to scare some people, but I bet that could be addressed with a mouseover that says something like, "Another type of u-shaped curve. A parabola is not the shape formed by a rope hanging from two ends..." or something similar. This kind of mouseover ought to make this section totally appropriate for the basic description. By the way, it's totally possible to have both a mouseover and a link on the word parabola.
A couple of details on "The Catenary Mathematically":
- The second equation should read "cosh(x)" on the left-hand-side of the equation, instead of just "cosh".
- You write that the two equations in this section are identical, which they aren't. What if you said something like, "the second equation defines the cosh symbol that is used in the first equation."
- Two notes on this sentence: It is important to note that catenaries are not a family of curves; the differences between catenaries comes from the scaling factor a from the equations and the value of x.
- I would think that cateneries are a family of curves, because each value of a gives a different curve.
- Any catenary takes every possible value of x as input, so the value of x does not actually give rise to different curves, and it's not even clear what "the value of x" refers to. You may be trying to say something that is right on; I'm just not clear what that is.
Just going by the sub heading "The Catenary and the Parabola Conceptually" I really do think that should be in the basic description. I think it will add a lot to the page to have that unhidden at the beginning. It's a really great explanation that doesn't need any math beyond algebra 1, which I would consider pretty basic.
The cosh totally belongs in the next section (let's face it, hyperbolic functions scare people!), but I'm afraid that people won't look at that simple explanation because they're too afraid it will require more than than they know.
I'd suggest you get another opinion on this (probably from Abram), too, because I see you haven't changed it from before. If you could at least give me a solid explanation of why you feel it fits in the mathematical description, I'd really appreciate it.
First, check out the email I sent you about this page. But also, I'd move your discussion of the bridges up into "a basic description" and leave just the math in the "a more mathematical description."
Would you like me to go over how one can derive the catenary equation from the physics of the hanging string? You could also find it in a lot of physics books... try Classical Mechanics by John Taylor (this is the phys 111 text book).