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Reflection Matrix in 2D

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Here is a simple setup of a manipulation and reflection matrix in 2D space.
By using a reflection matrix, we can determine the coordinates of the point
P
1
, the reflected image of the point
P
in the line defined by the vector
L=(u,v)
from the origin.
The projection of
P
onto the line is
P
0
. The point
P
1
is then determined by extending the segment
PP
0
by
|
PP
0
|
. As vectors,
P
1
=2
P
0
-P
.
If
L
is normalized (so that
2
u
+
2
v
=1)
, the reflection matrix is
R=
1-2
2
v
2uv
2uv
-1+2
2
v
. Then
R·R=
I
2
, that is, the reflection of a reflection is the identity. Also,
det(R)=-1
.
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