Graphics Reference
In-Depth Information
Fig. 12.1
Rotating a point by
a double reflection
12.3 Reflecting a Vector
Figure
12.2
shows a mirror with a unit normal vector
n
and a vector
a
with its
ˆ
reflection
a
. Vector
a
has a perpendicular component
a
and a parallel component
⊥
n
, and our objective is to derive a definition of the reflection
a
in terms of
vector
a
and any other essential vectors.
From our knowledge of multivectors, we know that
a
with
ˆ
n
2
ˆ
=
1 which permits us to
write
n
2
a
a
= ˆ
= ˆ
n
(
na
).
ˆ
This has created the geometric product
na
which equals
ˆ
na
ˆ
= ˆ
n
·
a
+ ˆ
n
∧
a
(12.1)
therefore,
a
= ˆ
n
(
n
ˆ
·
a
+ ˆ
n
∧
a
).
(12.2)
ˆ
ˆ
ˆ
ˆ
We can see that (
12.2
) has two parts:
n
(
n
·
a
)
and
n
(
n
∧
a
)
. The first part is another
way of expressing
a
:
a
=
(
n
ˆ
·
a
)
n
ˆ
and as
a
=
a
⊥
+
a
Fig. 12.2
Reflecting a vector
in a mirror