Graphics Reference
In-Depth Information
AB
AB
BC
BC
and
¢¢
¢¢
are probably not the same.
To find the parallel projection T of
R
3
2.4.1.1. Example.
onto the plane
X
defined
by the equation
xyz
-+=
2
3
parallel to
v
= (3,1,1).
Solution.
Clearly, given a point
p
, if t is chosen so that
p
+ t
v
belongs to
X
, then
T(
p
) =
p
+ t
v
. Let
p
= (x,y,z). Solving
(
)
-+
(
)
++
(
)
=
xt
+
32
yt
zt
3
for t, gives t = (1/2) (-x + 2y - z). It follows that T is defined by the equations
1
2
3
2
9
2
x
¢ =-+-+
x
3
y
z
1
2
1
2
3
2
y
¢ =-+-+
x
2
y
z
1
2
1
2
3
2
.
z
¢ =-+++
x
y
z
A parallel projection between two hyperplanes in
R
n
2.4.1.2. Theorem.
preserves
parallelism, concurrence, betweenness, and the ratio of division.
Proof.
Easy.
2.4.1.3. Theorem.
Any map of the plane onto itself that is a composition of
parallel projections is an affine map. Conversely, every affine map in the plane is a
composite of parallel projections.
Sketch of proof.
The first statement follows from the fact that lines are preserved.
Now let T be an affine map. Assume that
A
,
B
, and
C
are noncollinear points with
T(
A
,
B
,
C
) = (
A
¢,
B
¢,
C
¢). First, project
R
2
to a plane
X
that contains
A
and
B
so that
C
gets sent to a point
C
1
. Next, project
X
back to
R
2
in such a way as to send
C
1
to
C
¢.
It follows that the composite of these two projections sends
A
to
A
,
B
to
B
, and
C
to
C
¢. Repeat this process on
A
¢ and
B
¢. See Figure 2.24.
The construction in the proof of Theorem 2.4.1.3 shows that any affine map can
be realized as a composite of at most six projections.
