A plane touches the globe at the north pole. A line through the south pole and through another point on the globe intersects that plane. That intersection point is the image, under the stereographic projection, of that other point on the globe than the south pole. The image of the south pole is the point at infinity in the one-point compactification of the plane.
It has long been known (for more than 2000 years, I think?) that
- this mapping takes circles on the globe to circles on the plane, and
- this mapping is conformal.
So my question is: What sorts of converses to this theorem are there?
And a secondary question: What is the history of this theorem and its converses and their proofs?
Answer to these questions can be found in the Wiki reference.
Since the projection mapping is bijective, circles on the plane ( of north -pole) map to circles on the globe when mapped from south pole as center for projection. An oblique cone with vertex at south pole cuts the sphere and a plane along these circles.
It is a conformal mapping. Angles are conserved but sense of rotation is reversed. Mapping of complex variables are all conformal.
If we plot the circles in Geogebra etc. it would be instructive. Begin with plotting of arbitrary curves by inversion in the plane ( plot them both) and then only proceed to stereographic 3D projection plotting. From similar triangles it is possible to derive all the transformation (mapping) formulas.
Latitudes/longitudes map as a polar grid and vice-versa.
However if you rotate the entire globe by $90^{\circ}$ maintaining tangent contact at a point on equator now making it new center of projection you obtain bipolar co-ordinate grid of orthogonal circles from lat/long circles... on a perpendicular plane containing both poles and center of sphere. The unit circle length appears in this projection interestingly at several places (power of circles, distance to concurrent points that are north/south pole ) images. This is also bijective.
Let $\angle PNS = \theta$
From similar triangles $ NPS, NSP'$
$$ \rho = P'N=2a \sec \theta,\quad R= PN= 2a \cos \theta $$
$$ \rightarrow \rho \cdot R = 4 a^2 = 1,$$
if diameter $2a =1 $ for all $\theta.$
The projection holds for any figure on the sphere onto the plane.
To understand the phenomenon of projection the underlying bilinear Moebius complex variable mapping where circles map to circles is to be studied at first in 2D for effects of magnification, distortion and shift of the circles mapped: See e.g., at 31.00 in Balakrishnan Lecture
$$ w=\frac{az+b}{cz+d} $$
There is also an invariant Cross-ratio in the mapping.
Apart from this, fixed point referred to the given and mapped circles in stereographic projection can be identified as the north ( or south ) pole of sphere as the common vertex of oblique cone projection.
Wiki states that the projection was known from the first century, mostly due to need of astronomical body position studies. Mentioned in the modern form in Riemann's famous Habilitation lecture about Foundations of geometry.