Graphics Reference
In-Depth Information
We finish the chapter with several special topics. Section 3.8 discusses a generalization
of the usual central projection. Section 3.9 describes the beautiful theorem of Pascal
and some applications. The last topic of the chapter is the stereographic projection.
Section 3.10 describes some of its main properties.
3.2
Central Projections and Perspectivities
Definition.
Let
O
be a fixed point of
R
n
. For every point
p
of
R
n
distinct from
O
, let
L
p
denote the line through
O
and
p
. If
Y
is a hyperplane in
R
n
not containing
O
, then
define a map
n
p
o
:
RY
Æ
by
()
=«
=
p
O
pL Y L
, if
intersects
Y
in a single point,
p
p
undefined, otherwise.
The map p
O
is called the
central projection with center
O
of
R
n
to the plane
Y
. If
X
is
another hyperplane in
R
n
, then the restriction of p
O
to
X
, p
O
|
X
:
X
Æ
Y
, is called the
perspective transformation
or
perspectivity from
X
to
Y
with center
O
.
Note that our terminology makes a slight distinction between central projections
and perspectivities. Both send points to a plane, but the former is defined on all of
Euclidean space, whereas the latter is only defined on a plane; however, they clearly
are closely related.
Clearly, from the point of view of formulas, one would not expect our new maps
to be complicated because they simply involve finding the intersection of a line with
a hyperplane. Let us look at some simple examples to get a feel for what geometric
properties these maps possess. First, consider perspectivities between lines in
R
2
.
Figure 3.1 shows the case where the two lines
parallel
. In this case, the ratio of the
distance between points and the distance between their images is constant. The per-
spectivity is one-to-one and onto. It preserves parallelism, concurrence, ratio of divi-
sion, and betweenness.
What happens when the two lines are
not
parallel? See Figure 3.2. The point
V
on
L
has no image and the point
W
on
L
¢ has no preimage. These points are called
vanishing points
. Betweenness is
not
preserved as is demonstrated by the points
A
,
B
, and
C
in Figure 3.2. Furthermore, the fact that betweenness is not preserved leads
to other properties not being preserved. In particular, segments, rays, and ratios of
division are not preserved, and distances are distorted by different constants.
Next, consider perspectivities between planes. When the planes are parallel, things
behave pretty well just like for parallel lines. The interesting case is when the planes
are not parallel. Consider a perspectivity with center
O
from a plane
X
, which we shall
call the object plane, to another plane
Y
, which we shall call the view plane. The fol-
lowing facts are noteworthy.
