text: The first stereographic projection defined in the preceding section sends the "south pole" (0, 0, −1) of the unit sphere to (0, 0), the equator to the unit circle, the southern hemisphere to the region inside the circle, and the northern hemisphere to the region outside the circle.
  The projection is not defined at the projection point  = (0, 0, 1). Small neighborhoods of this point are sent to subsets of the plane far away from (0, 0). The closer  is to (0, 0, 1), the more distant its image is from (0, 0) in the plane. For this reason it is common to speak of (0, 0, 1) as mapping to "infinity" in the plane, and of the sphere as completing the plane by adding a point at infinity. This notion finds utility in projective geometry and complex analysis. On a merely topological level, it illustrates how the sphere is homeomorphic to the one-point compactification of the plane.
  In Cartesian coordinates a point  on the sphere and its image  on the plane either both are rational points or none of them:
  Stereographic projection is conformal, meaning that it preserves the angles at which curves cross each other (see figures). On the other hand, stereographic projection does not preserve area; in general, the area of a region of the sphere does not equal the area of its projection onto the plane. The area element is given in  coordinates by
  Along the unit circle, where , there is no inflation of area in the limit, giving a scale factor of 1. Near (0, 0) areas are inflated by a factor of 4, and near infinity areas are inflated by arbitrarily small factors.
  The metric is given in  coordinates by
  and is the unique formula found in Bernhard Riemanns Habilitationsschrift on the foundations of geometry, delivered at Göttingen in 1854, and entitled Über die Hypothesen welche der Geometrie zu Grunde liegen'.
  No map from the sphere to the plane can be both conformal and area-preserving. If it were, then it would be a local isometry and would preserve Gaussian curvature. The sphere and the plane have different Gaussian curvatures, so this is impossible.
  Circles on the sphere that do not pass through the point of projection are projected to circles on the plane. Circles on the sphere that do pass through the point of projection are projected to straight lines on the plane. These lines are sometimes thought of as circles through the point at infinity, or circles of infinite radius. These properties can be verified by using the expressions of  in terms of  given in : using these expressions for a substitution in the equation  of the plane containing a circle on the sphere, and clearing denominators, one gets the equation of a circle, that is, a second-degree equation with  as its quadratic part. The equation becomes linear if  that is, if the plane passes through the point of projection.
  All lines in the plane, when transformed to circles on the sphere by the inverse of stereographic projection, meet at the projection point. Parallel lines, which do not intersect in the plane, are transformed to circles tangent at projection point. Intersecting lines are transformed to circles that intersect transversally at two points in the sphere, one of which is the projection point. (Similar remarks hold about the real projective plane, but the intersection relationships are different there.)
  The loxodromes of the sphere map to curves on the plane of the form
  where the parameter  measures the "tightness" of the loxodrome. Thus loxodromes correspond to logarithmic spirals. These spirals intersect radial lines in the plane at equal angles, just as the loxodromes intersect meridians on the sphere at equal angles. 
  The stereographic projection relates to the plane inversion in a simple way. Let  and  be two points on the sphere with projections  and  on the plane. Then  and  are inversive images of each other in the image of the equatorial circle if and only if  and  are reflections of each other in the equatorial plane.
  In other words, if:
  *  is a point on the sphere, but not a north pole  and not its antipode, the south pole ,
  *  is the image of  in a stereographic projection with the projection point  and
  *  is the image of  in a stereographic projection with the projection point ,
  then  and  are inversive images of each other in the unit circle.