From Math Images
Checklist for Writing Pages
Added by Greg Brown.
Messages to the Future
- Very long page. There is not much more, if anything, to add.
- One possible addition is an explanation under Property 5 justifying the images that use Cartesian rectangles, since the proof relies on the principle carrying over to polar and spherical rectangles. The principle does, in fact, carry over, but it is not referred to right now. Diana provided me some feedback as to how this can be done, but since the page has been down I never added it and probably will not have time now.
- More could potentially be added to the Why It's Interesting section, but the Cartography content (and page in general) is very long right now.
- The Complex Analysis could theoretically be expanded, but it's meant to be a short overview; the Riemann Sphere page would build off of stereographic projection if anyone were interested in taking the topic further.
Reference and Footnotes
- Reference for photography content.
- All images cited when appropriate.
- Otherwise, no content in need of citation.
- The usefulness of stereographic projection is clear.
- Reference to Complex Analysis (and the uncommon operations we're allowed to do in the extended complex plane).
- Extensive content on cartography, which I would say is pretty interesting.
Quality of Prose and Page Structure
- I think the Basic Description and main image are compelling enough for readers who are not too advanced in math. The language is kept relatively simple, but the page gets the point across. It's also a neat way to introduce mapping to a basic audience.
- Once you get to the More Mathematical Section, the reader is not really spared of any technical language, but that is, I think, appropriate. The Definition section lumps a lot of things to remember in one place, but it's really necessary for this page to have the conventions clearly established.
- I am very happy with the writing quality on this page. I think it's readable, deliberate, and clear in most places.
Integration of Images and Text
- Images are well-integrated. Many proofs are accompanied with images to make them clearer.
- Images include both those that establish mathematical relationships and those that illustrate the point (the proofs of Properties 2 and 5 particularly exemplify this).
- Very consistently reference to images in-text. Images are all anchored and numbered.
Connection to Other Mathematical Topics
- Generally it was not necessary to link to other math topics. I think the only case was spherical coordinates, which I actually explain in a lot of depth on the stereographic projection page since it's so important.
Examples, Calculations, Applications, Proofs
- Lots of proofs. Every equation and property is proven.
- Main application included is Cartography, which isn't really presented in a mathy way.
- In hindsight, there are not many examples, ie. I never say we have such-and-such point on the sphere and it projects to such-and-such point on the plane. However, I don't think such examples are really necessary on this page. The equations are used more for the proofs and general conceptual understanding of stereographic projection.
Mathematical Accuracy and Precision of Language
- I think most of the language is appropriate.
- Fairly accessible. Some technical language in More Mathematical Section.
- I try not to use complex notation. For example, I don't use set notation in the Definition section since it would be needlessly complicated. The exceptions are in some of the proofs, like that for Property 2. For the difficult proofs, I take the opportunity to introduce set notation (although I explain it when I do so). The language usage corresponds with how "deep" the reader is in the page.
- When I get back to the Cartography section I make the language slightly more colloquial.
- Paragraph size is good.
- Sequence of page is logical. Presents definition, equations, properties, and complex analysis, before moving onto Cartography as the main application.
- Each section is more or less necessary for the others, with some exception where reasonable (Cartography more or less stands alone, or with minimal background in the More Mathematical Explanation; complex analysis doesn't really require having gone through the Properties section but would not make sense before it).
- Appropriate use of bold-face and balloons for introducing/defining terms.
- Hiding where appropriate, all formatted right.
- There are some large paragraphs, particularly in the Cartography section. However the paragraph break-up all makes sense, and they aren't that long, so I don't think it's an issue.
- The page works for smaller window sizes. The Mercator/Gall-Peters projections alongside each other are a little big, but I don't think it would stretch on most computers.
Figure 1: Point A appears to be on the sphere's cross-section and is mapped onto the x-axis at Point B. The other line has no labels. We talked about using specific references in the explanatory text and moving this Figure below Figure 2, making a smoother transition from 3D to 2D.
Chris 4:00, 26 June 2013 (EDT)
More Mathematical Explanation
Chris 4:00, 26 June 2013 (EDT)
- P2: Change "and some point Q on the plane." The idea is that the point does not already exist; it is created as P is mapped onto the plane.
- P4, beginning with "Let's restate...",
- S3: Change "and some point Q" to "and mapped onto Q"
- S5: Preferred spelling is "collinear."
Projection in terms of spherical coordinates
Spacing issue with Figure 3 (when you zoom in, there is a large blank space). There are also spacing issues for the coordinates for R a couple paragraphs later.
Chris 4:00, 26 June 2013 (EDT) Proof #1: You might consider both writing (x/1-z)^2 + (y/1-z)^2. Also, add an explanation for some of your steps, for instance when you substitute 1 for x^2 + y^2 + z^2.
I don't get how an oblique cone's intersection with a plane forms a circle and not an ellipse.
Here we will also prove this theorem analytically. What theorem?
Define W. Show labels in Figure 5.
Why It's Interesting
Chris 4:00, 26 June 2013 (EDT)
- P1, last sentence: Change "left us" to "created"
- P2: Why does locally mean near the pole opposite the projection?
- Change "upper hemisphere" to "Northern hemisphere." "The red circle" is hard to distinguish.
- We talked in the meeting about the words "accurate" and "inaccurate." Instead, I would specify in which way the projection does or does not match the real world (area or conformality).
--Chris 11:30, 28 June 2013 (EDT) We talked in the meeting about the words "accurate" and "inaccurate." Instead, I would specify in which way the projection does or does not match the real world (area or conformality).
P5: Instead of "is no different", say it is "very similar to that for a polar, except that…"
P6: Connect this paragraph with the preceding one. z=0 is polar and z= 1 or -1 is "transverse." (I know you have a word to replace "transverse.")
It really works well to have the Mercator and Gall-Peters projections next to each other!
P8: To strengthen your point, give the actual ratio in size between Greenland and Africa.
As discussed, Figure C will make more sense if preceded by the three azimuthal projections applied to the Earth.
Comments from 2009
Anna 7/7/09 Can you make the animation bigger? I can't see what's going on :(
Sorry about the delay, but I finally fixed that bug, thanks for the heads up.
I really like what you've done with the applet. One bug I've noticed is that if you click and drag the mouse far enough off the applet, the applet itself blacks out. It would be great if you could fix this problem, but otherwise, it's not a huge deal. It's a great addition to the page!
Hey I've modified the applet to accept user input, but instead of using sliders you can simply click and drag your mouse to rotate the sphere: Stereographic Projection
Actually the applet does work for me, I just needed to update my java platform. That's a very nice animation, thanks! Based on the animation you've made, maybe some sliders could control the orientation of the sphere- a slider or two could rotate the sphere in different ways, such as one to rotate in the -direction and one to rotate in the -direction.
To be clear, direction would be rotation away from the x-axis within the x-y direction and direction would be rotation 'away from' the z-axis: -Brendan 6/26
This looks like a pretty interesting illustration of the concept YouTube Stereographic Projection.
Hey, the applet isn't loading from me.
I've created a basic stereographic projection applet, check it out and let me know what else should be in there (i'll get to the sliders soon): Stereographic Projection
-Steve (from Drexel) 6/24
Hey Drexel folks,
An applet for this page would be really cool. I was thinking of an applet with some sliders that would allow the user to move a point on the sphere around, and a line would actively show the user where the point they control is projected to on a plane.
Can you use the Riemann Sphere page as something related?
I added a link on your Riemann Sphere page back to this one, and added a link back to the Riemann Sphere page from this one. I didn't want to discuss the Riemann Sphere on this page, because I think it would overcomplicate this page.
The way you added that was exactly what I had in mind, and I think it works well.