Graphics Reference
In-Depth Information
This definition has the advantage that one does not need to know a normal vector for
the line (only a direction vector or a second point
B
on the line). Of course, finding a
normal vector to a line in the plane is trivial. On the other hand, our normal vector
definition of a reflection will generalize to higher dimensions later.
Finally, since
=+
(
)
Q
P
PA • N N
,
we see that
Q
is the point where the line through
P
that is orthogonal to
L
meets
L
.
Therefore, another definition of S(
P
) is that we solve for that point
Q
and then define
()
=+2.
S
PPPQ
(2.11)
To put it another way, the segment
PP
¢
is perpendicular to the line
L
and intersects
the line at its midpoint
Q
.
2.2.3.1. Theorem.
Let S be the reflection about a line
L
.
(1) The definition of S depends only on
L
and not on the point
A
and the normal
vector
N
that are chosen in the definition. The three definitions of a reflection
specified by equations (2.9), (2.10), and (2.11) are equivalent.
(2) If t is chosen so that
P
+ t
N
is the point where the line through
P
with direc-
tion vector
N
meets the line
L
, then S(
P
) =
P
+ 2t
N
.
(3) The fixed points of S are just the points on its axis
L
.
(4) If
L
is the axis of a reflection S and
L
¢
is a line orthogonal to
L
, then
S(
L
¢) =
L
¢.
(5) Reflections are motions.
Proof.
Exercise.
2.2.3.2. Example.
To find the reflection S
x
about the x-axis.
Solution.
If we choose
A
= (0,0) and
N
= (0,1), then
PA
=-
P
and
()
=+
()()
[
]
()
S
x
PP
2
P
•,
0 1
0 1
,,
or
(
)
=-
(
)
Sxy
,
x y
,
.
x
In other words, S
x
has equations
xx
y .
¢=
¢=-
(2.12)
2.2.3.3. Example.
To find the reflection S about the line
L
defined by the equation
2x - y + 2 = 0.
