Nonlinear Projections - Transformations and Projections in Computer Graphics - page 171

y

vertical
line

x

top of
barrel

vertical
line

R cos θ

R θ

v

line of sight

R φ

of observer

φ

θ

R

α

x y

z z

Rα

barrel

cut
here cut

here

P

(a) (b)

φ R cos θ

y

πR cos θ , R θ )

( πR cos θ , R θ )

( −

band for

large θ

R φ

x

( πR , 0 )

(0,0)

curved image of

vertical line

( πR cos θ , −R θ )

( −πR cos θ , −R θ )

φ R cos( − θ )

(c)

Figure 4.27: Spherical Panoramic Perspective.

barrel is now cut behind the observer and is unrolled into a flat, two-dimensional fig-
ure resembling a Band-Aid (Figure 4.27c) that's called a band or a capsule (see also
Figure 4.58). The image seen by the observer through the barrel is displayed on this
band, in contrast with the cylindrical panoramic projection, where the projected image
is displayed on a rectangle.
At its center, the band has a width of 2 πR (the circumference of the sphere),
while at the top and bottom its width equals 2 πR cos θ . The height of the band is 2 Rθ .
Truncating the sphere into a barrel makes it possible to control the amount of distortion
in the final projected image. Small values of θ result in a narrow band whose shape is
close to a rectangle. Only a small part of the scene around the observer is displayed
on this band, but with a minimum of distortion. When θ is set close to π/ 2, the band