Nonlinear Projections - Transformations and Projections in Computer Graphics - page 169

the left, cars are parked pointing toward us. (One such car can be seen at the extreme
right of the image.) These are the two directions of the same street. The center street,
where we see a park bench, stroller, and people walking, is a paved walkway sandwiched
between the two directions of the street. The implicit assumption behind this image is
that viewers' familiarity with street scenes will help them to “straighten out” the distor-
tions in the image and thus to enjoy it. The reader should also notice that vertical lines
in this image seem to tilt toward the edges of the image, and this tilting becomes more
pronounced for lines close to the edges. This is probably an artifact of the particular
software used to create these images. Part (c) of this figure shows a large space serving
as artists' studios in Lyon, France. Here we see the four sets of curved horizontal lines
that are the hallmark of Figure 4.23b. The vertical lines are also tilted as in part (b).
An intuitive way to understand and accept curved perspective is to print the curved
projection of a familiar scene on paper, roll the sheet of paper into a cylinder, go inside
into the center, and look around at the scene. (This may be simple if the projection
incorporates less than 360 ◦ .) When seen this way, any curves on the paper that are the
projections of straight lines should look straight. This method also provides a simple
test of any software used to compute and render the projection.
Commercial software for creating cylinder-shaped panoramas already exists. Pop-
ular examples are the Apple QuickTime VR Authoring Studio , PhotoVista from Live
Picture Inc., and PhotoStitch , which comes with every Canon digital camera. A qual-
itative discussion of curved perspective can be found in [Ernst 76], pp. 102-103. The
well-known drawing High and Low by M. C. Escher is an example of curved perspective.
4.6 Spherical Panoramic Projection
The following quotation, from [Ernst 76], suggests a way to generalize the cylindrical
panoramic projection of the previous section.
Perhaps it has already struck you that the cylinder perspective used by Escher,
leading to curved lines in place of the straight lines prescribed by traditional
perspective, could be developed even further. Why not a spherical picture
around the eye of the viewer instead of a cylindrical one? A fish-eye objective
produces scenes as they would appear on a spherical picture. Escher certainly
did give some thought to this, but he did not put the idea into practice, and
therefore we will not pursue this further.
The idea raised by Ernst (but not pursued by Escher) is to imagine a transparent
sphere placed around the observer, where everything seen by the observer through the
sphere is fused (or painted by the observer) onto the sphere's material. The sphere is
then somehow flattened, resulting in a full 360 ◦ spherical perspective. The trouble with
this idea is that a sphere cannot be unrolled into a flat surface without introducing
further distortions (see Section 4.14).
We start with what is perhaps the simplest approach to the problem of deforming
and flattening a sphere. Once a three-dimensional point P has been projected onto the
surface of the sphere, it becomes a point P ∗ with longitude and latitude. We construct