Graphics Reference
In-Depth Information
1.6
Orientation
This section is an introduction to the concept of orientation. Although this intuitive
concept is familiar to everyone, probably few people have thought about what it means
and how one could give a precise definition.
The notion of orientation manifests itself in many different contexts. In everyday
conversation one encounters phrases such as “to the left of,” “to the right of,” “clock-
wise,” or “counterclockwise.” Physicists talk about right- or left-handed coordinate
systems. In computer graphics, one may want to pick normals to a planar curve in a
consistent way so that they all, say, point “inside” the curve. See Figure 1.8. A similar
question might be asked for normals in the case of surfaces. How can one tell in a
systematic way that our choice of normals is “consistent”? What does this really mean?
Probably the easiest way to demonstrate the orientability property for surfaces is
in terms of the number of “sides” that they have. Consider the cylinder in Figure 1.9(a).
This surface has the property that if one were a bug, the only way to get from the
“outside” to the “inside” would be to crawl over the edge. We express this by saying
that the cylinder is “two-sided” or orientable. Now, a cylinder can be obtained from
a strip of paper by gluing the two ends together in the obvious way. If, on the other
hand, we take this same strip of paper and first give it a 180-degree twist before we
glue the ends together, then we will get what is called a
Moebius strip
(discovered by
A.F. Moebius and independently by J.B. Listing in 1858). See Figure 1.9(b). Although
p
3
p
5
p
4
N
5
N
3
p
6
N
4
N
2
p
2
N
6
N
1
p
1
p
7
Figure 1.8.
Uniformly oriented normals.
Meridian
Cylinder
Moebius Strip
Figure 1.9.
Induced orientations
along paths.
(a)
(b)
