The making of ...
The images here are created using stereographic projections,
it is one of the many ways of mapping points on a sphere onto a plane.
Since a sphere and a plane are topologically different forms, the mapping cannot be performed
without some form of distortion. Stereographic projections, also called planisphere projections
have been employed by Hipparchus and documented by Ptolemy, it arises as a way of mapping
spherical data onto an image plane in a range of fields that include astronomy,
cartography, geology, and mathematics.
The first stage is the creation of a spherical projection, otherwise known as a
equirectangular projection of a sphere. There are many ways of doing this photographically,
they are mostly distinguished from each other by the final image resolution. The simplest is
with a 180 degree fisheye lens and SLR camera. Three images are captured each separated
by 120 degrees horizontally.
Since these three images capture the entire visual field, 360 degrees horizontally and 180
degrees vertically, they can be stitched and blended together to form a spherical
projection, as follows.
It is this image, mapped as a texture onto a sphere, that can be projected onto a
plane using a stereographic projection.
A stereographic projection involves selecting a focal point (normally along a vertical line
through the origin) and a plane which will become the projected image plane.
To determine where any point on the sphere maps to on the image plane, a ray is drawn
from the focal point through the point in question. Where this ray intersects the image
plane is the projected position.
In this case the focal point is the north pole of the sphere and the image plane is
tangential to the sphere and touches the south pole. This is illustrated below for two points
in the image, the top of the building roof and the yacht mast.
Stereographic projections preserve angles (conformal) but they do not preserve lengths (obvious
if one considers what happens to points towards the north pole) and it therefore follows
that it does not preserve area (isometric). The projection is smooth (no discontinuities), at
least for points on the sphere between the focal point and the image plane.
