Reflection Matrix in 2D

-0.600	0.800
0.800	0.600

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

.