I don't get stereographic projection in hyperbolic geometry... Help.

[Al B. Sure!]

can anybody explain this to me?

First, we're told to consider a sphere S whose equator coincides with the bounding circle of a projective representation of a hyperbolic plane. The next part is very intuitive but to give you the situation, i'll type it:

"To pass from the projective representation in the plane (considered as horizontal) to the new one on the sphere, we simply project vertically upwards. The straight lines in the plane, representing hyperbolic stright lines, are represented on S by semi-circles meeting the equator orthogonally."

not difficult to visualize but here is where I'm lost:

"Now, to get from the representation on S to the conformal representation on the plane, we project from the south pole. This is what is called stereographic projection. Two important properties of stereographic projection are that it is conformal, so that it preserves angles, and that it sends circles on the sphere to circles (or, exceptionally, to straight lines) on the plane. Can you see how to prove these two properties? (Hint: Show, in the case of circles, that the cone of projection is intersected by two planes of exactly opposite tilt)."

blah?

[KrazyHorse]

Which part don't you understand? Do you not understand what the stereographic projection is at all?

[Al B. Sure!]

yeah. i dont get that at all. I also don't get those two properties but i might once i understand stereographic projection.

[yin26]

Every once in a while I tell myself: "Wasn't life better back in school?" Thank you, Albert, for reminding me of how awesome it felt to finish school and focus on much more interesting things (like annual reports, staff meetings and car taxes).

[KrazyHorse]

To project a point from the surface of the upper hemisphere (which is the domain we're dealing with) back to the plane which dissects the sphere, draw a line from the point to the south pole of the sphere (the pole not included in your domain). Where this line intersects the plane is your projected point.

[Al B. Sure!]

nevermind. i looked on wikipedia... it's just the same thing as when you convert a spherical globe to a flat map.... of course the circles that do not pass through the center will remain circles on the plane (ie- latitude lines) while circles that do pass through the center will be striaght lines (longitude).

i get it now. should've looked it up before coming here.

[Al B. Sure!]

yin:

Every once in a while I tell myself: "Wasn't life better back in school?" Thank you, Albert, for reminding me of how awesome it felt to finish school and focus on much more interesting things (like annual reports, staff meetings and car taxes).

school? It's summer break. I'm reading this for... fun


and thanks, KH. i get it now. it's a lot more obvious than it seemed when i read it.

[KrazyHorse]

Red dot is south pole. Green dot projects to orange dot. Blue dot projects to yellow dot

[Al B. Sure!]

or, the first picture here:

[KrazyHorse]

No. In that case, the plane of projection is tangential to the north pole. In this case the plane of projection bisects the sphere.

[TheStinger]

I can't believe you don't know this.

[Al B. Sure!]

but for all effective purposes, is it not the same? The points would end up in the same places?

[KrazyHorse]

No, of course not.

[KrazyHorse]

When the plane of projection is tangential at the north pole then the equator projects to a circle with twice the radius of the sphere. When the plane of projection bisects the sphere at the equator, the equator projects to a circle with the same radius as the sphere.

[Al B. Sure!]

A point 1 degree above the equator would be at the same place on the plane in both cases...