Nonlinear Projections - Transformations and Projections in Computer Graphics - page 183

4 Nonlinear Projections
The height S oftheopenconeisgivenby S = √ H 2 + R 2 . The vertica l angle β

between the xy plane and the direction of P is given by tan β = z/ x 2 + y 2 . (Notice
that β variesfrom0to90 ◦ .) Once β is known, the polar coordinate r is determined
by r = S (1 sin β ). It varies between 0 and S . The top angle α of the open cone is

computed from 2 R/S =cos( α/ 2), and the polar coordinate θ lies between 0 and α ,so
it is given by θ = αγ/ (2 π ), where γ is determined by the x and y coordinates of P by
means of tan γ = |y/x| (90, 180, or 270 degrees may have to be added to γ depending
on the quadrant, see Figure 4.21).

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4.10 Panoramic Cameras

A typical dictionary definition of panorama is “a picture taken in three-dimensional
space and presented on a continuous surface encircling the viewer.”
There are a large variety of lenses available for current cameras (both digital and
film based), ranging from extreme wide angle to powerful telephoto, but even the widest
wide-angle lenses cannot capture an image that spans more than 180 ◦ . Most fisheye
lenses can capture 180 ◦ images, but the result is highly distorted, especially along the
edges. Professional as well as amateur photographers like to be able to stand at a given
point and capture an image of everything visible from all directions, which explains
why panoramic cameras are popular. Inexpensive high-resolution digital cameras have
become powerful and popular, and this has encouraged the development of panoramic
software. Given a digital camera and a tripod, it is easy to take a series of overlapping
photos, input them directly from the camera into the computer, and stitch them by
software into a panorama (normally cylindrical). In spite of this, special panoramic
cameras, both digital and film-based, the latter of which have been made since the
1840s, are still being made and used.
An important resource for information on all aspects of panoramic cameras is the
International Association of Panoramic Photographers [IAPP 05], whose mission is “to
educate, promote, exchange artistic and technical ideas, and to expand public aware-
ness regarding panoramic photography.” Two important resources maintained by this
organization are a list, located at [cameraInproduction 05], of panoramic cameras in
production and a timeline of panoramic cameras, located at [cameraTimeline 05].
A fun guide for do-it-yourselfers is [funsci 05]. Information on panoramic cameras
and creating panoramic images can be found on many Internet sites. See, for example,
[shortcourses 05] and [philohome 05].
A new reference topic for this topic is [Jacobs 05].
There are currently three types of cameras that capture panoramic images: a rotat-
ing camera, a swing-lens camera, and a camera with a parabolic panoramic lens system.
The first two can produce undistorted images, while the third type produces a highly
distorted image that has to be “unfolded” by special software to look like other types
(normally cylindrical or cubic) of panoramas. A description of all three (followed by a
note on pinhole cameras) follows.