# Thanks!

Your matrix transpose animation is so useful, thank you! Toddintr (talk) 17:01, 18 January 2021 (UTC)

## Earth as a Manifold

I've been thinking a lot about visualizing different coordinate charts for S^2 by using Earth's surface as a frame of reference. This would help show the different kinds of distortions (changes in areas, angles, etc.) you get with the different coordinate systems. Some natural candidates would be:

• standard spherical coordinates
• stereoscopic projection
• cylindrical projection
• the exponential map
• as a graph, over the unit disk (of course this only applies to 1 hemisphere at a time, however)

The tricky part is taking the data from the Earth's surface, and mapping it to the different domains in R^2 through the inverse coordinate transformations; I don't even know where/how you would extract that kind of data about our planet's surface.

In my mind these animations would be interactive, allowing the user to pinch and pull and twist the planet around, showing the different ways these transformations change the look of the domain in R^2. For example, in the cylindrical example, you could position the globe any which way inside the cylinder, which would lead to all sorts of transverse Mercator projections. Or with the exponential map, you could move your anchor point all around the surface of the globe, and since geodesics emanating from the center are simply straight lines, you could choose any two points on the globe to see how much the line "bends" in the other charts. You could even get a sense for the way that the transitions between charts looks, by comparing one common coordinate patch in the two (or more) systems of coordinates.

Anyway, these are just a bunch of the ideas I've had floating around in my head these past couple of weeks, and I could think of no one better to share them with than you. Of course if these aren't things you have time or energy to think about right now, then by all means feel free to ignore them. I just thought it could be fun to collaborate on some more fun math together!

See you around (no pun intended)

P.S. One extra, fun/goofy idea I also had was to generate a "donut" shaped version of Earth. In my mind it would go something like this:

1. Take inverse spherical coordinates to map the Earth's surface onto the domain [0,2pi) x [0,pi).
2. Apply a linear transformation to stretch the domain to [0,2pi) x [0,2pi).
3. Apply an affine transformation so that the branch cuts in the domain occur entirely in water (so as to hide them, visually). This isn't so important in the theta direction, since the fundamental domains for the sphere and torus agree in this direction; but the fact that the torus is also periodic in the phi direction, whereas the sphere is not, needs to be addressed.
4. Apply a standard map from the domain [0,2pi) x [0,2pi) to R^3 to generate a torus (with some R > r).
5. Apply the Earth's "skin" onto this torus. — Preceding unsigned comment added by MathPatch (talk • contribs) 04:48, 1 February 2021 (UTC)

## Spindle torus render

Connection hidden by near rim

Hi LucasVB,

Many thanks for uploading the torus cross-sections in 2007. They help me visualise the latter two classes of torus well.

Is it possible to make a change to File:Standard_torus-spindle.png to show the connection of the spindle to the outer surface?

As moving the camera will make it inconsistent with the other images, unless you can rerender them as well, perhaps reducing the amount of overlap of the two circles will shorten the spindle enough to allow the connection to be visible above the near rim.

cmɢʟee ⋅τaʟκ 18:21, 25 May 2021 (UTC)

## Circle image using capital theta for 360 degrees

Lucas,

I came across your circle animation at https://boycottholland.wordpress.com/2014/07/14/an-archi-gif-compliments-to-lucas-v-barbosa/

Super awesome!

I also like the idea of capital theta for 360 degrees (along the lines of Michael Cool's "Trig Rerigged" paper of 2008).