Nonlinear Projections - Transformations and Projections in Computer Graphics - page 161

obvious (but perhaps not very useful) extension of circle inversion is sphere inversion,
where the spaces inside and outside a sphere are swapped. Reference [Coxeter 69]
presents the complete theory of circle inversions. A more general treatment of inversive
geometry can be found in [Stothers 05].
Figure 4.20 (after [Gardner 84]) shows the circle inversion of a chessboard.
Figure 4.20: The Circle Inversion of a Chessboard.
4.4 Panoramic Projections
Visitors to an exceptionally lovely spot sometimes wish they could see the view behind
them as well as in front of them simultaneously. This kind of effect is generated by the
various panoramic projections . A panorama is defined as an unbroken view of an entire
surrounding area, and panoramas have always been a favorite with artists, painters,
and photographers. The insert below discusses the Mesdag panorama, one of the few
surviving large panoramas painted in the 18th and 19th centuries. When cameras came
into general use in the early 20th century, inventors started developing panoramic cam-
eras (Section 4.10). With the advent of fast, inexpensive personal computers and digital
cameras in the 1980s, it became possible, even easy, to take a sequence of (partially over-
lapping) photographs with any camera and stitch them by software into a single picture
thatdepictsalargearea,sometimesanentire360 ◦ view around a point, including parts
that are very high or very low and cannot normally be included in a single picture.
The price for including so much visual information in one picture is distortion. Any
method for projecting a three-dimensional scene into a panoramic picture introduces
some distortion. Straight lines become curved and familiar shapes may look funny or
become completely unrecognizable.
The main types of panoramic projections described here are the cylindrical, spher-
ical, and cubic. All three are based on the same principle, but only the first is popular