![]() Thus, we have a class of transformations that keeps little planets looking like little planets, at least as long as we don’t tilt the sphere too far. You may recall that a great circle on the sphere that passes through the North Pole (a meridian circle) maps stereographically to a line through the center of the projected image. If we tilt the viewable sphere by 90 degrees, the horizon circle from the original scene becomes a meridian, and so projects to a line. Tilting by a little more or a little less than 90 degrees maps the horizon to a very large circle-it appears to be nearly a straight line when projected stereographically. ![]() The sky lies on one side and the ground on the other, much like we see in “ordinary” photographs. But in such a projection we enjoy nearly a full 360-degree field of view along the horizon (impossible in an ordinary photo!), and this implies that there must be some serious distortion. See figure 3 for an example.Īnd if we tilt the viewable sphere a full 180 degrees, so that it is entirely upside down before projecting, then the horizon maps to the same circle as in the original projection. ![]() But now it is a circle with the sky on the inside-resulting in a fantasticallooking tunnel world. While stereographic projection has been used in cartography for centuries, in the 19th century Bernhard Riemann used it to gain a critical insight into the complex numbers. If one associates the projection plane beneath the viewable sphere with the complex plane-so that the sphere has radius one and rests precisely on the origin-then stereographic projection provides a one-to-one correspondence between the complex numbers and the points on the sphere except for the North Pole. Said another way: if a single point were added to the set of complex numbers, this “extended complex plane” could be placed in perfect one-to-one correspondence with the points on a sphere via stereographic projection. The extra point is called the point at infinity, and the sphere in this context is known as the Riemann sphere. It so happens that the field of complex analysis, from Riemann’s time to the present, has produced a cornucopia of conformal mappings from the extended complex plane to itself. This gives the panoramic photographer a deep reservoir of possibilities for producing flat images from the viewable sphere. The new possibilities come about by identifying the viewable sphere with the extended complex plane via stereographic projection, and then applying conformal mappings from the extended complex plane to itself to obtain new images. In fact, the simple sphere-tilting operation described earlier can be described this way: it can be thought of as a Möbius transformation.
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