Just for fun, you can seduce a geometry off a flat plane onto the surface of a sphere that makes it act like a space with a singular point (my preferred term).
Let S2 be the surface of a sphere with radius 1. That provides the points and curves of the system. Let E2 be a flat two-dimensional plane, like an endless rug that S2 is sitting on. Set up 3D coordinates for the sphere S2 and its rug E2, letting the third coordinate be fixed at value 0 in defining the plane. Let point <0, 0, 0> be the one and only point shared by sphere and plane. The center of the sphere is at <0, 0, 1>, but this is not a point of either space. The top of the sphere (antipode to <0, 0, 0>) is <0, 0, 2>. This is a point of S2.
Here is how we make the special geometry for S2. For any point <x, y, z> of S2, draw a 3D line through <0, 0, 2> and that point <x, y, z>. That line will meet the plane E2 at one point <x', y', 0>. Likewise, for any point <x', y', 0> of E2, draw a line through <0, 0, 2> and <x', y', 0>. That will cross the sphere S2 at one different point <x, y, z> of S2. Therefore, there is a 1-1 mapping that carries points of S2 to points of E2, except for <0, 0, 2>. <0, 0, 2> is our singular point for the coming geometry on S2 and there is no line through it that meets E2 at some point without crossing through the sphere S2. This mapping is called a stereographic mapping of the sphere (minus one point) to the plane. It is invertible.
Here are some coordinate transformation equations for the stereographic mapping function:
mapping points of S2 to points of E2 --
x' = 2x/(2-z)
y' = 2y/(2-z)
z' = 0
(notice that this fails when z = 2, the singular point on the sphere)
mapping points of E2 back to points of S2 --
x = 4x'/(x'2 + y'2 + 4)
y = 4y'/(x'2 + y'2 + 4)
z = 2 - 8/(x'2 + y'2 + 4)
(notice that z is constrained to have a value between 0 and 2, except 2 itself is unallowed. This is because the denominator x'2 + y'2 + 4 is constrained to be value 4 or larger, so 8/(x'2 + y'2 + 4) is constrained to be value 0 to 2. But the only way this last term could have value 0 is if the denominator is not a finite value. Our points <x', y', 0> all have finite coordinates. So z must be 0-or-larger, up to, but NOT including, 2. The top point of the sphere is never reached.)
Alway remember that the points <x, y, z> of S2 are constrained to lie on the unit sphere, so the coordinates must always obey the sphere equation:
(x-0)2 + (y-0)2 + (z-1)2 = 12
. That just boils down to:
x2 + y2 + z2 = 2z
. The <x', y', 0> points are only constrained to have finite-valued coordinates, with z' always equal to 0.
The stereographic mapping allows us to map euclidean curves and figures from S2 to the flat space E2. We look at what the images of these subsets are in the 2D geometry of E2 and attribute these geometric attributes to the originals on S2. So, a straight line on S2 will be something that maps to a euclidean line on E2 and a circle of S2 will be something that maps to a euclidean circle on E2. The same rule holds for line segments, figures, etc. This is the stereographic geometry on S2.
It turns out that the circles of this geometry are those euclidean circles (not only so-called "great circles") on the sphere that do not cross point <0, 0, 2>. Also, the lines of this geometry are those euclidean circles on the sphere that DO cross point <0, 0, 2>. Remember that <0, 0, 2> itself is NOT mapped to anything by the stereographic mapping. This geometry is a non-euclidean geometry, since there are no truly parallel lines (lines always contain point <0, 0, 2>). <0, 0, 2> is the one and only singular point of this space.
