Graphics Reference
In-Depth Information
2.2.3
Reflections in the Plane
Another important type of motion is a reflection. Such a motion can be defined
in several ways. After giving our definition we shall discuss some of these other
characterizations.
Definition.
Let
L
be a line in the plane. Define a map S :
R
2
Æ
R
2
, called the
reflec-
tion about the line
L
, as follows: Choose a point
A
on
L
and a
unit
normal vector
N
for
L
. If
P
is any point in
R
2
, then
()
=¢ = +
(
)
S
PPP PA•NN
2
.
(2.9)
The line
L
is called the
axis
for the reflection S.
The reader will find Figure 2.4 helpful as we discuss the geometry behind reflec-
tions. First, note that
W
= (
PA
•
N)N
is just the orthogonal projection of the vector
PA
onto
N
. Define a point
Q
by the equation
==
(
)
PQ
W
PA • N N
.
Intuitively, it should be clear that
Q
is the point on
L
as shown in Figure 2.4. This
does not follow from the definition however and must be proved. The following string
of equalities:
(
)
=
(
[
)
]
AQ • N
=
PQ
+
AP
•
N
PA • N N
+
AP
•
N
=
PA • N
+
AP • N
=
0
shows that
Q
satisfies the point-normal form of the equation
AX
•
N
= 0 for the points
X
on the line (or hyperplane)
L
, so that
Q
does indeed lie on
L
. Furthermore, it is
easy to check that
AQ
is the orthogonal projection of
AP
on
L
. This means that, if
V
is a unit direction vector for
L
, then
AQ
= (
AP
•
V)V
and we could have defined the
reflection S by
()
=+
(
)
S
P
P
2
PA
+
AQ
.
(2.10)
P + tN
N
P
L
B
W = (PA·N)N
A + sAB
Q
A
P¢
Figure 2.4.
Defining a reflection in the
plane.
