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
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