The pole corresponds to the point at infinity, which results is the projective number line. You have to define some point as pole, since ##\mathbb{S}^1 \simeq SO(2,\mathbb{R}) \simeq U(1,\mathbb{C}) \simeq \mathbb{P}(1,\mathbb{R})## is the unit circle, which is not the same topological space as ##\mathbb{R}^1##.FallenApple said:So the circle minus the pole is homeomorphic to the number line. Does that mean that the pole itself represents a number that doesn't exist on the real line? After all, the pole certainly exists and yet is the only point that isn't mapped.
The "pole" in stereographic projection refers to the point on the surface of a sphere or globe where the projection is centered. It is also the point where all the meridians (lines of longitude) converge.
The pole is typically chosen to be the North or South pole of the sphere, depending on the desired orientation of the projection. It can also be determined by the location of a specific point on the surface of the sphere.
The pole is used to create a two-dimensional representation of the Earth's surface on a flat plane. It allows for accurate representation of the shape and relative size of land masses and bodies of water.
One limitation of stereographic projection is that it is not suitable for mapping large areas due to distortion towards the edges of the projection. Additionally, the pole must be chosen carefully in order to accurately represent the desired region on the map.
The center of projection is the point where the globe is projected onto the flat plane, while the pole is the point on the surface of the globe where the projection is centered. They are often the same point, but can be different depending on the desired orientation of the projection.