Nonlinear Projections - Transformations and Projections in Computer Graphics - page 174

horizontal lines located close to the center of the picture (noticeable in the upper half)
are essentially straight. The five-point grid of Figure 4.34 is an artist's tool that helps
draw such pictures. Reference [New Perspective 98] has more on such tools.
This section explains the principles behind the five-point grid. The material pre-
sented here is based on the concept of curvilinear perspective , developed by Albert
Flocon and Andre Barre [Flocon and Barre 68]. Curvilinear perspective is a two-step
spherical panoramic projection whereby points in the 180 ◦ space in front of the ob-
server are first projected on a hemisphere and then from the hemisphere onto a flat
circle. When this is repeated for the 180 ◦ space behind the observer, the result is two
circles that contain the entire 360 ◦ of space surrounding the observer.
Their topic beckons us to join with the fun and excitement, but it is also a revo-
lutionary manifesto, a call to liberation from dogma. Not “Down with Traditional
Perspective!” but “Down with the Tyranny of O cial Rules.” Not “Learn the Only
True Perspective!” but “Let a Hundred Flowers Bloom!”
—Robert Hansen in [Flocon and Barre 68]
Figure 4.29a illustrates the first step. A point P in space is projected to a point
P ∗ on a hemisphere. The observer is located at the center of the sphere. Part (b) of the
figure shows how the hemisphere is projected onto a flat circle. The center of the circle
is tangent to point R on the sphere (the point right in front of the observer). Given a
point Q on the sphere, we draw the great-circle arc from R to Q . Denoting the length
of this arc by L ,point Q is projected to the point at distance L from the center of the
circle in the direction from R to Q . This particular projection of a hemisphere to a
circle was proposed in the 16th century by Guillaume Postel and has the useful property
that its distortions of angles and distances are minimal. Clearly, the distance between
R and Q on the hemisphere is preserved on the circle, whereas the distance between
points A and B on the hemisphere of Figure 4.29c suffers a minimal distortion. For a
30 ◦ angle, the ratio between the arc length AB and its projection is only 1.01, and for
a90 ◦ angle this ratio is 1.57, much smaller than distance distortions caused by other
sphere projections.
Exercise 4.11: Show how to determine the distance between points A and B on the
hemisphere of Figure 4.29c and on the circle of the same figure. Compute the ratio of
these distances and show that it equals 1.01 for a 30 ◦ angle and 1.57 for a 90 ◦ angle.
Normally, the radius of the circle is R ( π/ 2) because this is the length of the longest
radial arc on a hemisphere of radius R . However, it is possible to extend the Postel
projection to project an arc of length r on the hemisphere to a segment of length sr on
the circle, where s is any desired scale factor. The radius of the circle in such a case is
sR ( π/ 2).
When the two steps of curvilinear perspective are performed for a vertical line,
it becomes a vertical curve on the circle (Figure 4.29d). This curve is very close to a
circular arc and for all practical purposes can be approximated by such an arc. Similarly,
a horizontal line in space is projected to a horizontal circular arc on the final circle. Lines
that are parallel to the line of sight of the observer are projected on the circle to straight
segments that converge at the center. Thus, the five-point grid of Figure 4.34 serves as