From Math Images
Everything basically looks great. The way you wrote, "the ratio of r to something" instead of the "ratio of these two numbers" takes care of that problem perfectly.
The mouse-overs on the conic sections in the chart are sweet.
I fixed two typos, one of which was replacing "it's focus" with "its focus".
The only other thing that this page could use is an explanation of what the angle in Kepler's equation refers to. Something like "position in its plane of orbit" or similar is fine
- Added a note explaining it's the standard theta in polar coordinates .
The numerical examples and the applications to orbits are both really nice. There is something of a slew of typos, many of which are things like typing "chart" as "cart", but I haven't gone in and fixed them because there are one or two sentences in there where I'm not quite sure how they should be fixed, so you might want to give these sections a re-read at some points.
- I've tried to find them. Let me know if I've missed anything.
The algebraic perspective should probably mention that the conic equation can be re-written through simple algebra in the forms presented in the chart, so that those equations that you make heavy use of in the numerical examples don't seem to come out of nowhere.
- Done .
I think I really don't like the line about how circles are by definition circular. The reasoning in that line is, well, very circular! I think you could strike that phrase without any problem.
The sentence where you replaced "of" with "if": you should specify what "these two numbers" are. Even though it's mathematically obvious, there's some rule against using a pronounish adjective or whatever you call words like "these" without an antecedent.
- Personally, on a math page, I'd rather have slightly incorrect grammar, than a much longer, clunkier sentence. Feel free to change it if you disagreed .
The internet lied to you ?!!! Unbelievable. You should let the New York Time, the Washington Post, and Jon Stewart know immediately. Also, we should fire whomever is in charge of the internet. I think that's Al Gore.
- I'll send an angry letter posthaste.
So, lots of small things.
First, I agree, strike that row on the chart, which also allows you to strike the definition of the variable "c" for hyperbolas.
Second, I killed the mouse-over on degenerate cases, because it literally just repeated the main text. I also slightly edited the locus definition of conic sections because the grammar wasn't exactly right (it's the locus of points, not the locus of a point), and you don't need to say the distance between "a point on d" to P -- you can just say the distance between P and d.
Third, flush out the orbit section a little bit more. Did somebody show that if the thing being orbited is stationary and if it and the orbiting object are the only massive objects in the universe, and newtonian mechanics is right, then orbits will exactly be conic sections? I know that's a lot of ifs, but written slightly differently, it would be really interesting.
- Kepler did! The proof, while not terrible, has too much physics in it to present on this page. The proof that his equation is the same as the conic section equation is a very, very painful proof that I haven't been able to find anywhere in full. Abbreviated, handwavy versions are 2+ textbook pages. Technically, both objects orbit the center of mass of the system and there are two equations (one for each body). For most planets in the solar system (I'm pretty sure Jupiter is the one exception) the center of mass of the system is actually inside the sun (suns mass 5000 times the earth's mass), so placing the sun at the center is a very good approximation. Even for Jupiter, I think it's off only by 1% or less.
- So my issue is how to flush this out without getting two physics heavy. I'll work on it, but it's an issue you should be aware of.
Other small things that could be changed are as follows.
The definition of axis of a cone is not helpful. I think we should strike it unless you want to write a better definition or even better replace with a picture.
- It now reads "The axis of a cone is a straight line that goes through the middle of the cone through the tip."
Somewhere we should mention in the text that we are dealing with a double cone.
- Added that into the first mouse over on "cone".
The description of ellipses as being formed by having plane of intersection that's greater than perpendicular to the axis is confusing, because the plane and the axis form two angles, one of which is obtuse and the other of which is acute. Can you think of how to reword this?
- Ok, now there are a million "of"s in the sentence, but I think it's better.
It took me a while to realize that two foci and two directrices are not necessary to define an ellipse or a hyperbola, but rather that there are two focus/directrix pairs that each generate the same ellipse/hyperbola. Can you be more clear about this up front?
- No, I won't be, because I realized it's not true!! I'm fixing that!! I did some math to check it, and it just doesn't work that way. The internet lied to me!!
The non-circularness interpretation of eccentricity is maybe too important to be hidden in mouse-over?
- Yes, fixed.
The line in the circle description about how there's a ratio that must be zero has a typo, but I can't figure out exactly what the words you intended are.
- of/if problem. Fixed
All of my new content is in. I'd like to know about the row in the chart I mentioned... I think I should delete it, but I'd like a second opinion.
So I'm editing away now. I'm trying to actually explain what focus and directrix are, and I'm
working on filling in the chart done filling in the chart. You can see that I've made some progress. My last bit will be adding in how all of this relates to orbits, just because I think that's cool. I'm thinking of fulling showing how to calculate the properties of the ellipse and hyperbola that I made pictures of, just because I've already taken the time to do it, and I don't think it will take more than an hour or so to type up.
One of my issues with the chart/table is that I have no idea what "relation to focus" is supposed to mean. At all. I vote for eliminating this row of the chart, do you agree?
My other main issue is that I actually can't seem to figure out how to make all of the math work out. I keep trying to pick an equation for a hyperbola, then calculate where the focus should be, and where the directrix should be, but it never looks right. I spent a solid hour and a half working on creating a pretty picture, until I realized that said pretty picture basically contradicted what I wanted it to say... so then I spent about hour failing to understand the algebra of this, which really shouldn't be that hard. I can't find a good source online... so I'm just going to keep on working away at this. Hopefully, there will be instructive pictures on this page soon.
I fixed it! I don't know what was wrong with me earlier today, but a second time around meant that I got a nice, pretty hyperbola picture with numbers that work
The "relation to focus" row of your chart is still a mystery to me.
"a^2 - b^2 = c^2". What does this have to do with the focus? Maybe the row just has a bad label...
"p = p". This is really cryptic. p always equals p. that has nothing to do with conic sections. do you mean p can be anything, but depends on the particular parabola?
"a^2 + b^2 = c^2". Again, what does this have to do with the focus?
Finally, the sentence before the chart needs some clarification. you refer to a conic being "centered at (0,0)". What does this mean for a parabola? Maybe you should write "centered at (0,0) (or, in the case of a parabola, with vertex at 0,0)". Also, for the hyperbola, parabola, and ellipse, you are not only assuming that the center is at (0,0). You are assuming a specific orientation.
Perhaps the chart needs a new row called "orientation". The circle entry can ready, "Center at (0,0)." The hyperbola entry can read, "opens vertically, center at (0,0)", etc.
I added the variable p to the circle section which should clarify this issue. Changing the rows and columns is very tedious and I don't think it would make that much of a difference. I also added a focus and directrix row and tried to fill it in as much as possible but not very successfully. Let me know what else is missing.
I've created the Conic section applet you requested: Conic Section
I agree switching the items would make the chart a bit clearer, but I don't feel like it would make that big a difference.
The main thing that confuses me in the chart is the row showing relation to focus. Is p a value, the coordinates of point, something else? From the earlier description, it sounded like P referred to the focus, so then maybe "p=0" should say "p=(0,0)".
Why does p appear in this row for some of the conic sections but not for others?
I believe that all this information is valuable -- I'm just not sure exactly what you're saying.
Also, your definition of conic section refers to a focus and a directrix, so there should be a row in the chart about the directrix (if you can't fill it in completely, fill in the part you understand and leave the rest for others).
I basically like the page and the chart. It's just this one part that confuses me. And I really want to know about things like the directrix of a circle.
I still think that switching the horizontal and vertical items in your chart would make it much more readable. Why don't you ask Abram for a second opinion?
In your basic description, you should link to your parabola page.
You bold "degenerate cases" but then don't explain what that means. Maybe use a mouse over?
Can you have a labeled diagram in your formal definition, so that we can see what all of those things look like?
In your table, I might actually switch your vertical and horizontal items.
Can you have a section how one figures out eccentricity?
If an applet could be designed where the viewer can choose where and what shape to cut a cone in and once the conic section is created, the name appears.