Talk:Inversion

Contents

Abram 8/5

Hey MaeBeth,

Nice job with the edits you have done. I agree with Anna's comments, except that linking to the Riemann Sphere may actually not make sense, because there's no need to invoke complex numbers to talk about this. You could link to stereographic projection, though the problem there is that stereographic projection doesn't just give a point at infinity: it also imposes a new metric on the plane, where points very far away from the origin are suddenly near each other. Does this metric tend to be associated with inversion?

The main things I would like to see changed are left over comments from before. If you just haven't been able to get around to them yet, no problem, but in case it's helpful, I've copied the remaining issues below.

• Some of the equations can stay in the more mathematical section, but easily understood consequences of these equations can be put in the more basic description. These consequences include statements like, "The inversion of a point always lies on the line connecting that point to the center of the circle of inversion", "Points that are very near the center of the circle have inversions that are very far away from the center of the circle", etc.
• Encourage people to notice these kinds of patterns in the animation

It would also be great if non-math folks could have a better idea what makes inversion interesting to mathematicians. A couple of ideas:

• You mention that it can simplify certain proofs. Can you expand on that a little bit?
• Describe in detail how inversion gives you a way of imagining a point at infinity that can be thought of as being extremely close to points that are way out there in every direction. I think some people will find that idea really cool, especially if you tie this to its mathematical usefulness.

Anna 8/3

Hi MaeBeth, I've got a few more suggestions:

This "We can now see that triangle OP'Q is similar to triangle OQP" is clear to me, but it takes a second to see, and anyone who doesn't remember what a similar triangle is (there are a lot of those people out there) are going to have a lot of trouble with that sentence just hanging out there.

In the next like, could you use the \frac{}{} command to make this "(OP)/k=k/(OP')" look more like a ratio? I think that makes it easier to see, and it's a very small fix.

You mention "the point at infinity" without first clarifying what you mean. You might consider linking to Riemann Sphere which explains one way to do that... You need some justification for why you can say that there is a point at infinity.

I also feel a bit iffy with "Solving the equation k/(OP')=0 where k is a constant" because it's not really "solving" if the result is infinity... can you pick out another word?

In your inversion of curves section, it would be really helpful to have pictures of what you're talking about next to each little paragraph.

Anna 7/29

I agree with Abram about moving stuff up to the basic description, and describing the patterns more thoroughly in text next to the demo.

I also strongly agree with his last paragraph.

One slightly picky thing. This sentence "For example, Q is a point on the circle such that OQ is perpendicular to PQ, and P' is the foot of the altitude of the triangle OQP where QP' is perpendicular to OP" has too much information. Can you break it up a bit?

Abram 7/22

This page is great. I'm going to make a lot of comments, but it's only because I've been doing lots of these think-alouds so I've been exposed to a whole universe of things that can be confusing or upsetting from readers. Also, you do revisions really well, so I don't worry about giving you large tasks. With that in mind, here we go.

It would be great if this page could be made more accessible for non-mathematical people. Here are things that could make that happen:

• Move almost everything, including the animations, to the more basic description. If you worry this makes the basic description too long, it can start out hidden.
• Some of the equations can stay in the more mathematical section, but easily understood consequences of these equations can be put in the more basic description. These consequences include statements like, "The inversion of a point always lies on the line connecting that point to the center of the circle of inversion", "Points that are very near the center of the circle have inversions that are very far away from the center of the circle", etc.
• Encourage people to notice these kinds of patterns in the animation

It would also be great if non-math folks could have a better idea what makes inversion interesting to mathematicians. A couple of ideas:

• You mention that it can simplify certain proofs. Can you expand on that a little bit?
• Describe in detail how inversion gives you a way of imagining a point at infinity that can be thought of as being extremely close to points that are way out there in every direction. I think some people will find that idea really cool, especially if you tie this to its mathematical usefulness.

These two issues are the most important, but if you have more time, read on.

One thing that I think could make this page more accessible for everybody would be to replace some of the words in inversions of curves with symbols? For instance, label the center of the circle of inversion "O", and replace "Hence, we notice that inversion of a circle that passes through the center of the circle of inversion will always be a line that does not pass through the center of the circle of inversion" with "The inversion of a circle that passes through O is a line that does not pass through O." Static images of some of these properties could also be helpful.

MaeBeth 6/26

Just to let you know what I'm thinking about at the moment, I'm putting the image on hold for now because one of the Drexel students is working on creating a few things specifically for this page that will hopefully spice things up a bit. I'll definitely try to look more into what I can write more about the main image though. Thanks for the helpful comments!!

Brendan 6/26

The 'inversion of a point' section is very clear and easy to understand.

I'd agree that more simple images would be helpful, especially towards the end of the page. The sentences " Hence, we notice that inversion of a circle that passes through the center of the circle of inversion will always be a line that does not pass through the center of the circle of inversion. The inversion of a line not passing through the center of the circle of inversion is always a circle that does pass through the center of inversion" are calling for some clarifying pictures.

I'd also recommend discussing what's going on in the main image a little more.

Gene 6/19

"inversion of circles" I think is usually spoken of as "inversion in circles".

Both your Inversion and the Apollonius images are very nice!

It would sure be nice to have an interactive image--one where you could change P', or maybe one where you could take a figure (a segment or square or whatever) and look at it under inversion. Maybe with Drexel folks?

You've squished things too close. Important Properties shouldn't come until after the Inversion of a Point image. And the important properties merit their own diagrams with highlighted stuff.

Inversion of Curves--more simple images, please!

Good start, Maebeth.