Graphics Reference
In-Depth Information
points float around in
R
4
and only worry about projecting back to the “real” world at
the end. There will be no problems as long as we deal with individual points. Prob-
lems can arise though as soon as we deal with nondiscrete sets.
In affine geometry, segments, for example, are completely determined by their end-
points and one can maintain complete information about a segment simply by keeping
track of its endpoints. More generally, in affine geometry, the boundary of objects
usually determines a well-defined “inside,” and once we know what has happened to
the boundary we know what happened to its “inside.” A circle in the plane divides the
plane into two parts, the “inside,” which is the bounded part, and the “outside,” which
is the unbounded part. This is not the case in projective geometry, where it is not
always clear what is “inside” or “outside” of a set. Analogies with the circle and sphere
make this point clearer. Two points on a circle divide the circle into two curvilinear
segments. Which is the “inside” of the two points? A circle divides a sphere into two
curvilinear disks. Which is the “interior” of the circle?
Here is how one can get into trouble when one uses homogeneous coordinates
with segments. Again, consider Figure 4.20 and the segment corresponding to the
“real” points
A
and
B
. The figure shows that at least with some choices of represen-
tatives, namely,
p
1
and
p
2
, nothing strange happens. The segment [
p
1
,
p
2
] in
R
4
proj-
ects onto the segment [
A
,
B
] and so the points
s
pp
1
+
t
2
0
,
£
s t
,
£
1
,
s
+
t
=
1
,
represent the same points of
P
3
as the points of [
A
,
B
]. It would appear as if one can
deal with segments in projective space by simply using the ordinary 4-tuple segments
in
R
4
. But what if we used
p
1
¢=a
p
1
instead, where a < 0? See Figure 4.21. In that
case, the segment [
p
1
¢,
p
2
] projects to the exterior segment on
A
and
B
and so deter-
mines different points in
P
3
from [
A
,
B
]. The only way to avoid this problem would
be to ensure that the w-coordinate of all the points of our objects always stayed pos-
itive as they got mapped around. Unfortunately, this is not always feasible.
Figure 4.21.
Problems with homogeneous
representatives for points.
