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In fact, the simple sphere-tilting operation described earlier can be described this way: it can be thought of as a Möbius transformation. 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. This gives the panoramic photographer a deep reservoir of possibilities for producing flat images from the viewable 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.
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The extra point is called the point at infinity, and the sphere in this context is known as the Riemann sphere. 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. 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. 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. But now it is a circle with the sky on the inside-resulting in a fantasticallooking tunnel world.
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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 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. The sky lies on one side and the ground on the other, much like we see in “ordinary” photographs. 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. 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. 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. 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. Rotating the image of, say, a building to and from that light bulb at the top makes the building appear larger and smaller in the projection on the floor, in much the same way that the shadow of your hand in front of a light bulb would grow and shrink. This tilting operation is an excellent compositional tool, for it allows the panoramic photographer to emphasize the interesting parts and deemphasize the boring bits. In particular, the circular image of the horizon remains circular. It turns out this sphere-tilting operation has two useful properties: it preserves conformality, and circles that were in the original stereographic projection, before tilting, still appear as circles after the transformation.
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See figure 1.Įxploring this translucent ball idea further, we might ask what the panorama would look like if we were to tilt the ball (while keeping the light bulb directly above it, opposite the floor). The result is a striking image that looks like a little planet floating in the sky. We finished part I by describing a particular conformal mapping: the stereographic projection, which entails treating the viewable sphere as a translucent ball with a light bulb on top, from which the imagery is projected onto the floor. Conformal mappings work well because although there may be some large-scale distortions, small details do not get skewed or stretched into an unrecognizable mess. A certain class of these mappings, or projections, is well suited to displaying visible spheres: conformal mappings-functions that preserve angles when mapping imagery from one domain to another. To display spherical panoramas on a flat surface, we borrow ideas from cartography, the science of mapping a sphere to a plane. These panoramas capture the viewable sphere, an imaginary ball around the camera with imagery on it that matches the surrounding scene. Not just front, back, left, and right, but also up and down. Conformal Is the New Normal In part I of this series, we described photographic panoramas that capture the camera’s entire surroundings.