Graphics Reference
In-Depth Information
Proof.
Exercise.
2.2.3.7. Example.
To find the equation for the reflection S about the line in Example
2.2.3.3 using this approach.
Solution.
First translate the line
L
to a line
L
¢ that passes through the origin via the
translation
T: x = x +1
y=y
¢
¢
Next, let R be the rotation about the origin through the angle -q defined by
1
5
cos
q
=
,
0
££
q
p
2
.
R will rotate
L
¢ into the x-axis because q is the angle that the line
L
makes with the
x-axis. The equations for R and R
-1
are
1
5
2
5
:x=
1
5
2
5
-
1
R
:x=
¢
x
+
y
R
¢
x
-
y
2
5
1
5
2
5
1
5
y
¢=-
x
+
y
y
¢=
x
+
y
.
Finally, if S
x
is the reflection about the x-axis, then S is just the composite T
-1
R
-1
S
x
RT.
Since the equations for all the maps are known, it is now easy to determine the equa-
tions for S and they will again turn out to be the same as the ones as equations (2.13).
The reader might wonder at this point why we bothered to describe the solution
in Example 2.2.3.7 since it is more complicated than the one in Example 2.2.3.3. In
this instance, the method of Example 2.2.3.7 should simply be considered to be a case
of trying to give the reader more insight into how to solve a geometric problem. The
approach might not be efficient here but will be in other situations. It is important to
realize that there are two types of complexity: one, where we dealing with something
that is intellectually difficult, and the other, which may take a lot of time but only
involves intellectually simple steps. This is the case with the solution in Example
2.2.3.7. Actually, this type of question will probably come up again later on in this
chapter because there are usually many ways to solve problems. Any
particular
problem may very well have an extremely elegant solution that a human might find.
On the other hand, a computer is not able to deal with problems on a case-to-case
basis and needs a systematic approach.
2.2.4
Motions Preserve the Dot Product
2.2.4.1. Theorem.
If M is a motion with the property that M(
0
) =
0
, then M is a
linear transformation, that is,
